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Precise Estimates for Differences of the Gaussian Hypergeometric Function
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
215, 212]234 Ž1997.
AY975641
Precise Estimates for Differences of the Gaussian Hypergeometric Function G. D. Anderson* Department of Mathematics, Michigan State Uni¨ ersity, East Lansing, Michigan 48824
S.-L. Qiu Hangzhou Institute of Electronics Engineering (HIEE), Hangzhou, 310037, People’s Republic of China
and M. Vuorinen† Department of Mathematics, Uni¨ ersity of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014, Helsinki, Finland Submitted by Bruce C. Berndt Received July 15, 1996
By showing certain combinations of the Gaussian hypergeometric functions F Ž a, b; a q b; 1 y x c . and F Ž a y d , b q d ; a q b; 1 y x d . to be monotone on Ž0, 1. for given a, b, c, d g Ž0, `., a F b, and c - d, the authors study the problem of comparing these two functions. They find sup d g Ž0, a. : F Ž a, b; a q b; 1 y x c . F Ž a y d , b q d ; a q b; 1 y x d . for x g Ž0, 1.4, thus solving an open problem and a recent conjecture. Q 1997 Academic Press
1. INTRODUCTION In this paper we consider the Gaussian hypergeometric function F Ž a, b; c; x . s2 F1 Ž a, b; c; x . '
`
Ý ns0
* E-mail address: [email protected]. † E-mail address: [email protected]. 212 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
Ž a, n . Ž b, n . x n n! Ž c, n .
Ž 1.1.
213
PRECISE ESTIMATES FOR DIFFERENCES
for x g Žy1, 1., where Ž a, n. denotes the shifted factorial function Ž a, n. ' aŽ a q 1. ??? Ž a q n y 1., n s 1, 2, . . . , and Ž a, 0. s 1 for a / 0. In studying connections between the arithmetic-geometric mean and other mean values, J. M. Borwein and P. B. Borwein w9, Ž2.9.x showed that F
ž
1 1 1 1 , ; 1; 1 y x 2 - F y d , q d ; 1; 1 y x 3 2 2 2 2
/ ž
/
Ž 1.2.
for d s 1r6 and all x g Ž0, 1.. In w4, p. 79x it was conjectured that Ž1.2. holds for all d g Ž0, 1r6x and all x g Ž0, 1.. In w3, Theorem 4.3x it was proved that F
ž
1 1 1 1 , ; 1; 1 y x c - F y d 0 , q d 0 ; 1; 1 y x d 2 2 2 2
/ ž ž -F
1 2
yd ,
1 2
/
qd ; 1; 1 y x d - F
/ ž
1 1 , ; 1; 1 y x d 2 2 Ž 1.3.
/
for all d g Ž0, d 0 ., x g Ž0, 1., and c, d g Ž0, `. with 0 - 4 c - p d - `, where d 0 s ŽŽ dp y 4 c .rŽ4p d ..1r2 . It was conjectured in w3, Conjecture 4.10Ž1.x that for c s 2, d s 3 the best value of d 0 for which Ž1.3. is valid is
d0 s
1 2p
ž
p y 2 arcsin
2 3
/
s
1
p
arccos
2 3
f 0.268.
It is natural to ask what is the best value of d 0 s d 0 Ž c, d ., for which Ž1.3. holds for all x g Ž0, 1. and 0 - c - d - `. For the more general case, the following open problem is of interest. 1.4. Open Problem w4, p. 79, Open Problem Ž2.x. It is true, for small values of d , say 0 - d - min a, b4 , that F Ž a, b; a q b; 1 y x c . - F Ž a y d , b q d ; a q b; 1 y x d .
Ž 1.5.
for x g Ž0, 1., a, b, c, d g Ž0, `. with 0 - c - d - `? Considering the special case mentioned above, we can naturally raise the following two questions. 1.6. QUESTIONS. Ž1. For gi¨ en a, b, c, d g Ž0, `., c - d, what is the best ¨ alue of d 0 such that Ž1.5. is ¨ alid for all d g Ž0, d 0 x and all x g Ž0, 1.? Ž2. What can we conclude about the comparison between two functions in Ž1.5. for d 0 - d - min a, b4 if we can find d 0 in Question Ž1.?
214
ANDERSON, QIU, AND VUORINEN
In the present paper, we study Questions 1.6Ž1. and 1.6Ž2.. We shall not only find the full solution to Question 1.6Ž1., but also obtain error estimates for the difference of the two functions appearing in Ž1.5. whenever they are directly comparable for all x g Ž0, 1. Žsee Theorem 1.7 below.. In particular, our results imply the truth of w3, Conjecture 4.10Ž1.x for more general c and d Žsee Corollary 1.19. and give the answer to Open Problem 1.4. We shall also answer Question 1.6Ž2. in the case abc F d Žsee Theorem 1.13 below.. We now state our main results. 1.7. THEOREM. Let a, b, c, d g Ž0, `., a F b, p s a q b, c - d, and let d 1 be the unique solution for d of the equation cB Ž a y d , b q d . s dB Ž a, b . ,
d ) 0,
Ž 1.8.
where B Ž x, y . is the beta function, and let d 2 denote the unique positi¨ e root of the quadratic equation
d 2 q Ž b y a . d y ab 1 y
ž
c d
/
s 0,
that is,
d2 s
1
½
2
2 Ž b y a. q 4 ab 1 y
ž
c d
1r2
/
5
y Ž b y a. .
Then we ha¨ e: Ž1.
For 0 - d F d 1 , the function
hŽ x . '
1 1 y xc
F Ž a y d , b q d ; p; 1 y x d . y F Ž a, b; p; 1 y x c .
is strictly decreasing from Ž0, 1. onto Ž A1 , A 2 ., where A1 s A1 Ž d . '
A2
d cp
ab 1 y
½ ž
c d
y d 2 q Ž b y a. d
¡`, R Ž a, b . ¢B Ž a y d , b q d . y B Ž a, b . ,
s A Ž d . '~ R Ž a y d , b q d . 2
/
5
) 0,
Ž 1.9.
if 0 - d - d 1 , if d s d 1 ,
Ž 1.10.
215
PRECISE ESTIMATES FOR DIFFERENCES
while RŽ a, b . is defined by Ž2.4.. In particular, for all x g Ž0, 1. and 0 d F d 1, F Ž a, b; p; 1 y x c . q Ž 1 y x c . A1 - F Ž a y d , b q d ; p; 1 y x d . - F Ž a, b; p; 1 y x c . q Ž 1 y x c . A 2 ,
Ž 1.11. F Ž a, b; p; 1 y x c . q Ž 1 y x c . A1 - F Ž a y d 1 , b q d 1 ; p; 1 y x d . F F Ž a y d , b q d ; p; 1 y x d . - F Ž a, b; p; 1 y x d . , Ž 1.12. with equality in the second inequality in Ž1.12. if and only if d s d 1. Ž2. If d 1 - d - d 2 then, as functions of x, F Ž a y d , b q d ; p; 1 y x d . and F Ž a, b; p; 1 y x c . are not directly comparable on Ž0, 1., that is, neither F Ž a, b; p; 1 y x c . F F Ž a y d , b q d ; p; 1 y x d . nor its re¨ ersed inequality holds for all x g Ž0, 1.. In our next main result we study the case d 2 F d - min a, b4 . 1.13. THEOREM. Let a, b, c, d, p, d 1 , and d 2 be as in Theorem 1.7, and abc F d. Then, for d 2 F d - a, the function hŽ x . '
1 log Ž 1rx .
F Ž a y d , b q d ; p; 1 y x d . y F Ž a, b; p; 1 y x c .
is strictly increasing from Ž0, 1. onto Ž A 3 , A 4 ., where A3 s A3 Ž d . '
d BŽ a y d , b q d .
c
y
B Ž a, b .
Ž 1.14.
and A4 s A4 Ž d . ' y
d p
d 2 q Ž b y a . d y ab 1 y
ž
c d
/
F 0. Ž 1.15.
In particular, for all x g Ž0, 1. and d g w d 2 , a., F Ž a, b; p; 1 y x c . q A 3 Ž d . log
1 x
- F Ž a y d , b q d ; p; 1 y x d . - F Ž a, b; p; 1 y x c . q A 4 Ž d . log
1 x
- F Ž a, b; p; 1 y x c . , Ž 1.16.
216
ANDERSON, QIU, AND VUORINEN
F Ž a y d , b q d ; p; 1 y x d . F F Ž a y d 2 , b q d 2 ; p; 1 y x d . - F Ž a, b; p; 1 y x c . q A 4 Ž d 2 . log
1 x
s F Ž a, b; p; 1 y x c . , Ž 1.17.
with equality in the first inequality in Ž1.17. if and only if d s d 2 . 1.18. Remark. By symmetry of the function F Ž a, b; a q b; x . with respect to the parameters a and b, without loss of generality we may assume that a F b for convenience, as we did in Theorems 1.7 and 1.13. 1.19. COROLLARY. Ž1. For Ž1rp .arccosŽ crd ., the function
hŽ x . '
1 1yx
F
c
ž
1 2
y d,
1 2
0 - c - d - ` and 0 - d F d 1 '
q d ; 1; 1 y x d y F
/ ž
1 1 , ; 1; 1 y x c 2 2
/
is strictly decreasing from Ž0, 1. onto Ž A1 , A 2 ., where
A1 s A1 Ž d . s
A2
d 1 c 4
ž
1y
c d
y d 2 ) 0,
/
¡`, c ¢p ž d R y log 16 / ,
if 0 - d - d 1 ,
s A Ž d . s~ 1 2
Ž 1.20.
if d s d 1 ,
Ž 1.21.
and R s RŽ 12 y d , 12 q d ., while RŽ a, b . is defined by Ž2.4.. In particular, for all x g Ž0, 1., 0 - c - d - `, and 0 - d F d 1 ,
F
ž
1 1 , ; 1; 1 y x c q A1 Ž d . Ž 1 y x c . 2 2
/
-F -F
ž ž
1 2
y d1 ,
1 2
q d 1 ; 1; 1 y x d
/
1 1 1 c , ; 1; 1 y x c q R y log 16 Ž 1 y x c . 2 2 p d
/ ž
/
Ž 1.22.
217
PRECISE ESTIMATES FOR DIFFERENCES
and F
ž
1 1 , ; 1; 1 y x c q A1 Ž d 1 . Ž 1 y x c . 2 2
/
-F -F
ž ž
1
y d1 ,
2
1 2
q d 1 ; 1; 1 y x d F F
/ ž
1 2
yd,
1 2
q d ; 1; 1 y x d
1 1 , ; 1; 1 y x d , 2 2
/
/
Ž 1.23.
with equality if and only if d s d 1. Ž2. For 0 - c - d - ` and 12 1 y Ž crd . ' d 2 F d - 12 , the function
'
HŽ x. '
1
F
log Ž 1rx .
ž
1 2
yd,
1 2
q d ; 1; 1 y x d y F
/ ž
1 1 , ; 1; 1 y x c 2 2
/
is strictly increasing from Ž0, 1. onto Ž A 3 , A 4 ., where d
A3 s A3 Ž d . s
pž
A4 s A4 Ž d . s d
cos dp y
1 4
ž
1y
c d
/
c d
/
,
Ž 1.24.
y d 2 F 0.
Ž 1.25.
In particular, for all x g Ž0, 1., 0 - c - d - `, and d 2 F d F 12 , F
ž
1 1 1 1 1 , ; 1; 1 y x c q A 3 Ž d . log - F y d , q d ; 1; 1 y x d 2 2 x 2 2
/
-F
F
ž
1 2
yd, FF -F
ž ž
1 2 1 2
q d ; 1; 1 y x d y d2 ,
1 2
ž ž
/
1 1 1 , ; 1; 1 y x c q A 4 Ž d . log , 2 2 x Ž 1.26.
/
/
q d 2 ; 1; 1 y x d
/
1 1 1 1 1 , ; 1; 1 y x c q A 4 Ž d 2 . log s F , ; 1; 1 y x c . 2 2 x 2 2 Ž 1.27.
/
ž
Equality holds in the first inequality in Ž1.27. if and only if d s d 2 .
/
218
ANDERSON, QIU, AND VUORINEN
0 - c - d - `, if Ž1r p .arccosŽ crd . ' d 1 - d - d 2 ' then, as functions of x, F Ž 12 , 12 ; 1; 1 y x c . and F Ž 12 y d , d. q d ; 1; 1 y x are not directly comparable on Ž0, 1., that is, neither Ž3 .
1 2 1 2
For
'1 y Ž crd . , F
ž
1 1 1 1 , ; 1; 1 y x c - F y d , q d ; 1; 1 y x d 2 2 2 2
/ ž
/
nor its re¨ ersed inequality holds for all x g Ž0, 1.. Proofs of Theorems 1.7 and 1.13 and Corollary 1.19 will be supplied in Section 3. These theorems imply that, for a, b, c, d g Ž0, `., c - d, sup d : Ž 1.5. holds for all x g Ž 0, 1 . 4 s d 1 , where d 1 is as in Theorem 1.7. Furthermore, they give the correct asymptotic behavior of, and provide error estimates for, the difference of the two functions in Ž1.5. whenever they are directly comparable for all x g Ž0, 1..
2. PRELIMINARY RESULTS Before we prove our main results stated in Section 1, we need to establish several technical lemmas. But first, let us recall some known results for F Ž a, b; c; x . and for the gamma function. One of the tools we shall need in our work is the following asymptotic limit w12, Example 18, p. 299; 10, Theorem 1.10x, valid for c - a q b, limy Ž 1 y x .
aq byc
xª1
F Ž a, b; c; x . s
GŽ c. GŽ a q b y c. G Ž a. G Ž b .
,
Ž 2.1.
where G denotes the classical gamma function. As illustrated by Ž2.1., properties of the hypergeometric function are closely associated with the gamma function, its logarithmic derivative C Ž x ., and the beta function B Ž x, y .. For positive x and y these functions are defined by GŽ x. s
`
yt xy1
H0 e
t
dt, C Ž x . s
G9 Ž x . GŽ x.
, B Ž x, y . s
GŽ x. GŽ y. GŽ x q y.
Ž 2.2.
Žcf. w12x.. We shall need the Euler]Mascheroni constant g defined by n
g s lim
nª`
ž
Ý ks1
1 k
/
y log n s 0.57721 . . . ,
Ž 2.3.
219
PRECISE ESTIMATES FOR DIFFERENCES
and the function R Ž a, b . ' y2g y C Ž a . y C Ž b . .
Ž 2.4.
By w1, 6.3.3x, we have R
1 1 1 , s y2 g q C 2 2 2
ž
ž /
ž //
s log 16.
Ž 2.5.
The gamma function satisfies the difference equation w12, p. 237x G Ž x q 1 . s xG Ž x .
Ž 2.6.
if x is not 0 or a negative integer. It has the so-called reflection property w12, p. 239x GŽ x. GŽ1 y x. s
p sin p x
Ž 2.7.
if x is not an integer. We shall also need Stirling’s well-known asymptotic formula lim e x x 1r2yx G Ž x . s '2p
xª`
Ž 2.8.
w12, p. 251x. For other properties of these functions the reader is referred to w1, 6]8x. 2.9. LEMMA. a, the sequence
For a, b, c, d g Ž0, `., a F b, and c - d, and for 0 - d -
QŽ n. '
G Ž n qa yd . G Ž n qb qd . Ž a yd . Ž b qd . q Ž crd . Ž n qa qb . G Ž n qa . G Ž n qb . Ž n qabq a q b .
,
for n g N, is strictly decreasing, with lim n ª` QŽ n. s crd. Here, and in the sequel, N denotes the set of natural numbers. Proof. Let p s a q b, u s a y d , ¨ s b q d , and let Q1 Ž n . s Ž n2 q pn q u¨ . u¨ q
c d
Ž n q p q 1 . Ž n q ab q p .
y Ž n2 q pn q ab . u¨ q
c d
Ž n q p . Ž n q ab q p q 1 .
220
ANDERSON, QIU, AND VUORINEN
for n g N. Since, by Ž2.6., Q Ž n q 1. QŽ n.
s
Ž n2 q pn q u¨ .
u¨ q
Ž n2 q pn q ab .
u¨ q
c d c d
Ž n q p q 1 . Ž n q ab q p . Ž n q p . Ž n q ab q p q 1 .
for n g N, QŽ n. is strictly decreasing if and only if Q1Ž n. - 0 for all n g N. By elementary computation we obtain Q1 Ž n . s u¨
ž
c d
c
½
y 1 n2 q Ž u¨ y ab . u¨ q
/
Ž ab q p q 1 .
d
qpu¨ q Ž u¨ y ab . Ž ab q p . u¨ q
ž
cp d
/
q
cp d
ž
c d
y1
q abu¨
ž
/5 c d
n y1 ,
/
which is negative for all n g N since c - d and u¨ s ab y d Ž d q b y a. F ab. From Ž2.8., it follows that lim Q Ž n . s
nª`
s
c d
G Ž n q a. G Ž n q b .
nª`
c d
G Ž n q u. G Ž n q ¨ .
lim
lim
nª`
ž
= 1q
ž
s 2.10. LEMMA.
c d
nqu nqa
ay 1r2
/
ž
u¨ y ab n2 q Cn q ab
nq¨ nqb
by1r2
/
ž
nq¨ nqu
d
/
n
/
.
For a, b, u, ¨ g Ž0, `. with a q b s u q ¨ , and for n g N,
let QŽ n. s
G Ž n q u. G Ž n q ¨ . G Ž n q a. G Ž n q b .
.
Ž1. If u¨ s ab, then QŽ n. s 1 for all n g N. Ž2. If u¨ - ab Ž u¨ ) ab., then QŽ n. is strictly decreasing Ž increasing ., and lim n ª` QŽ n. s 1.
221
PRECISE ESTIMATES FOR DIFFERENCES
Proof. By Ž2.6., we have Q Ž n q 1. QŽ n.
s1q
u¨ y ab n q Ž a q b . n q ab 2
,
which clearly implies part Ž1. and the monotoneity of QŽ n. in part Ž2.. It follows from Ž2.8. that lim Q Ž n .
nª`
s lim
nª`
Ž n q u.
nq uy1r2
Žnq¨.
nq¨ y1 r2
Ž n q a.
nq ay1r2
Ž n q b.
nqby1r2
s lim 1 q nª`
n
u¨ y ab n2 q Ž a q b . n q ab
ž
nqu nqa
a
/ž
nqu nqb
b
/ž
nq¨ nqu
¨
/
s 1. 2.11. LEMMA.
For a, b g Ž0, `. with a F b, the functions
f 1Ž x . ' G Ž a y x . G Ž b q x . ,
f 2 Ž x . ' B Ž a y x, b q x . ,
and f 3 Ž x . ' R Ž a y x, b q x . are all strictly increasing and con¨ ex on Ž0, a., where R is as in Ž2.4.. Proof. Since C Ž x . s yg y
1 x
q
`
x
Ý ns1 n Ž n q x .
,
C9 Ž x . s
`
Ý ns0
1
Ž n q x.
2
Ž 2.12.
Žcf. w2, p. 198x. we see that C is strictly increasing on Ž0, `., while C9 is strictly decreasing there. Logarithmic differentiation gives f 1X Ž x . s f 1 Ž x . C Ž b q x . y C Ž a y x . , which is positive and, further, a product of two positive and strictly increasing functions on Ž0, a.. Hence the result for f 1 follows. Since f 2 Ž x . s f 1Ž x .rG Ž a q b ., the assertion about f 2 follows. By differentiation, we obtain f 3X Ž x . s C9 Ž a y x . y C9 Ž b q x . , which is positive and strictly increasing on Ž0, a..
222
ANDERSON, QIU, AND VUORINEN
2.13. LEMMA. Ž1. For n g N and c g Ž0, `., as a function of a, f nŽ a. ' C Ž a q n. y C Ž c y a q n. y C Ž a. q C Ž c y a. is strictly decreasing on Ž0, c . with f nŽ cr2. s 0. Ž2. For c g Ž0, `. and x g Ž0, 1., as a function of a, hŽ a. ' F Ž a, c y a; c; x . is strictly increasing on Ž0, cr2x, and decreasing on w cr2, c .. Proof. Ž1. Since C9Ž x . is strictly decreasing on Ž0, `. by Ž2.12., we get f nX Ž a . s C9 Ž a q n . y C9 Ž a . q C9 Ž c y a q n . y C9 Ž c y a . - 0, and hence the monotoneity of f n follows. Clearly, f nŽ cr2. s 0. Ž2. Since g n Ž a . ' Ž a, n . Ž c y a, n . s
G Ž a q n. G Ž c y a q n. G Ž a. G Ž c y a.
,
logarithmic differentiation gives g Xn Ž a . s g n Ž a . f n Ž a . ,
n g N,
so that, by Ž1.1., h9 Ž a . s
`
g n Ž a.
Ý Ž c, n . n! f n Ž a. x n . ns0
Hence part Ž2. follows from part Ž1.. 2.14. LEMMA.
For a, b g Ž1, `. with Ž a y 1.Ž b y 1. F 1, the function
h Ž x . ' Ž 1 y x . F Ž a, b; a q b y 1; x . y
ab aqby1
2 Ž 1 y x . F Ž a q 1; b q 1; a q b; x .
is strictly increasing and con¨ ex from Ž0, 1. onto Ž1 y abrŽ a q b y 1., 0.. Proof. Clearly hŽ0. s 1 y abrŽ a q b y 1., while Ž2.1. and Ž2.6. yield h Ž 1y . s
G Ž a q b y 1. G Ž 1. G Ž a. G Ž b .
y
ab
G Ž a q b . G Ž 2.
a q b y 1 G Ž a q 1. G Ž b q 1.
s 0.
PRECISE ESTIMATES FOR DIFFERENCES
223
Using the series expansion of F Ž a, b; c; x ., we get
hŽ x . s Ž 1 y x .
`
Ž a, n . Ž b, n .
Ý Ž a q b y 1, n . n! x n ns0 yŽ 1 y x .
s Ž1 y x.
`
`
Ž a, n q 1 . Ž b, n q 1 .
Ž a, n . Ž b, n .
xn Ý ns0 Ž a q b y 1, n . n! y
q
`
xn Ý ns0 Ž a q b y 1, n q 1 . n!
`
Ž a, n q 1 . Ž b, n q 1 .
xn Ý ns0 Ž a q b y 1, n q 1 . n!
n Ž a, n . Ž b, n .
xn Ý ns0 Ž a q b y 1, n . n!
s Ž 1 y x . Ž a q b y 1 y ab . s Ž x y 1.
`
Ý ns0
s Ž x y 1.
`
Ý ns0
s
`
`
Ý ns0
s
`
Ž a y 1, n q 1 . Ž b y 1, n q 1 . n x Ž a q b y 1, n q 1 . n!
Ž a q b y 1, n . n!
ns0
Ž a, n . Ž b, n .
Ž a y 1 . Ž b y 1 . Ž a, n . Ž b, n . n x Ž a q b y 1, n q 1 . n!
n Ž a y 1, n . Ž b y 1, n .
Ý
y
`
xn Ý ns0 Ž a q b y 1, n q 1 . n!
xn
Ž a y 1, n q 1 . Ž b y 1, n q 1 . n x Ž a q b y 1, n q 1 . n!
Ž a y 1, n . Ž b y 1, n .
Ý ns0 Ž a q b y 1, n q 1 . n!
nŽ a q b y 1 q n.
yŽ a y 1 q n. Ž b y 1 q n. x n s
`
Ž a y 1, n . Ž b y 1, n .
Ý ns0 Ž a q b y 1, n q 1 . n!
n y Ž a y 1. Ž b y 1. x n .
Since Ž a y 1.Ž b y 1. F 1, the monotoneity and convexity of h now follow.
224
ANDERSON, QIU, AND VUORINEN
2.15. LEMMA. For a, b, c, d g Ž0, `. with a F b and c - d, let d 1 and d 2 be defined as in Theorem 1.7. Then
d 1 - d 2 - a.
Ž 2.16.
Proof. Since the function hŽ x . ' x 2 q Ž b y a. x y abŽ1 y Ž crd .. is strictly increasing on the real axis with hŽ d 2 . s 0 s hŽ a. y abcrd, the second inequality in Ž2.16. holds, and the first inequality in Ž2.16. holds if and only if hŽ d1 . - 0
Ž 2.17.
for all a, b, c, d g Ž0, `. with a F b and c - d. Since, by Ž1.8. and Ž2.2., c d
s
B Ž a, b . B Ž a y d1 , b q d1 .
s
G Ž a. G Ž b . G Ž a y d1 . G Ž b q d1 .
,
Ž2.17. holds if and only if G Ž a y d 1 . G Ž b q d 1 . d 12 q Ž b y a . d 1 y ab - yabG Ž a . G Ž b . . Ž 2.18. Since d 12 q Ž b y a. d 1 y ab s yŽ a y d 1 .Ž b q d 1 ., from Ž2.6. we see that Ž2.18. can be written as G Ž a q 1 y d 1 . G Ž b q 1 q d 1 . ) G Ž a q 1. G Ž b q 1. , which is true by Lemma 2.11. For our final lemma we recall a monotone version of l’Hopital’s Rule w5, ˆ x Lemma 2.2 that will be useful in proving the quotient of two functions to be monotone. 2.19. LEMMA. For y` - a - b - ` let f, g : w a, b x ª R be continuous functions, differentiable on Ž a, b ., such that g 9Ž x . / 0 for x g Ž a, b .. If f 9Ž x .rg 9Ž x . is Ž strictly . increasing Ž or Ž strictly . decreasing . on Ž a, b ., then so is Ž f Ž x . y f Ž c ..rŽ g Ž x . y g Ž c .., where c s a or b.
3. PROOFS OF THE MAIN RESULTS In this section we prove Theorems 1.7 and 1.13 and Corollary 1.19. 3.1. Proof of Theorem 1.7. Let u s a y d , ¨ s b q d s p y u, and t s 1 y Ž1 y x . d r c. We note first, by Lemma 2.11, that d 1 is uniquely determined by Ž1.8., and that d 1 , d 2 satisfy Ž2.16., by Lemma 2.15.
225
PRECISE ESTIMATES FOR DIFFERENCES
For part Ž1. we let h1 Ž x . ' h Ž Ž 1 y x .
1rc
1
.s
x
F Ž u, ¨ ; p; t . y F Ž a, b; p; x . . Ž 3.2.
Let f Ž x . s F Ž u, ¨ ; p; t . y F Ž a, b; p; x ., and g Ž x . s x. Then g Ž0. s 0, fXŽ x. g 9Ž x .
u¨ d
s f 9Ž x . s
cp y
Ž1 y x.
ab p
Ž drc .y1
f Ž0. s
F Ž u q 1, ¨ q 1; p q 1; t .
F Ž a q 1, b q 1; p q 1; x . ,
Ž 3.3.
and f0 Ž x. s
u¨ d cp q y
ž
1y
d c
/Ž
1 y x.
Ž drc .y2
u¨ Ž u q1 . Ž ¨ q1 . d 2 p Ž p q1 . c
2
ab Ž a q 1 . Ž b q 1 . p Ž p q 1.
F Ž u q 1, ¨ q 1; p q 1; t .
Ž 1 yx . 2
wŽ drc .y1 x
F Ž u q2, ¨ q 2; p q 2; t .
F Ž a q 2, b q 2; p q 2; x . .
Ž 3.4.
We wish to show that h1Ž x . is strictly increasing on Ž0, 1.. Since h1Ž x . s f Ž x .rg Ž x ., the desired monotoneity will follow from Lemma 2.19 if we prove that f 0 Ž x . ) 0 on Ž0, 1. so that f 9Ž x .rg 9Ž x . s f 9Ž x . is increasing there. By expressing the hypergeometric functions in Ž3.4. in terms of power series and comparing coefficients, we are able to prove that f 0 Ž x . is indeed positive. The limiting values of h will be determined separately, by means of Lemma 2.11 and l’Hopital’s Rule. ˆ Let h2 Ž t . s a
u¨ d cp
q
ž
1y
d c
/Ž
1 y t . F Ž u q 1, ¨ q 1; p q 1; t .
u¨ Ž u q 1 . Ž ¨ q 1 . d 2 p Ž p q 1. c
2
2 Ž 1 y t . F Ž u q 2, ¨ q 2; p q 2; t . .
Then it follows from Ž3.4. that 2 Ž1 y x. f 0 Ž x.
s h2 Ž t . y
ab Ž a q 1 . Ž b q 1 . p Ž p q 1.
2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . .
Ž 3.5.
226
ANDERSON, QIU, AND VUORINEN
Using the series expansion for F Ž a, b; c; x ., we get h2 Ž t . s
2
d
Ž1 y t .
ž / c
ž
c d
y1
`
/ÝŽ ns0
qŽ 1 y t .
`
Ý ns0
s
2
d
Ž1 y t .
ž / c
`
Ý ns0
u, n q 1 . Ž ¨ , n q 1 .
Ž p, n q 1 . n!
tn
Ž u, n q 2 . Ž ¨ , n q 2 . n t Ž p, n q 2 . n!
Ž u, n q 1 . Ž ¨ , n q 1 . Ž p, n q 2 . n!
ž
c d
y 1 Ž p q n q 1.
/
q Ž u q n q 1. Ž ¨ q n q 1. y n Ž p q n q 1. t n s
d
2
ž / c
Ž1 y t .
`
Ý ns0
c Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q Ž n q p q 1 . t n , d Ž p, n q 2 . n! Ž 3.6.
while 2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . s F Ž a, b; p q 2; x .
Ž 3.7.
by w11, Theorem 21, p. 60x. From Ž3.7., we see that Ž1 y x . 2 F Ž a q 2, b q 2; p q 2; x . is strictly increasing in x on Ž0, 1., while trx s w1 y Ž1 y x . d r c xrx is strictly decreasing in x from Ž0, 1. onto Ž1, drc., by Lemma 2.19. Thus, since t ) x, the monotoneity of Ž3.7. gives 2 2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . - Ž 1 y t . F Ž a q 2, b q 2; p q 2; t . . Ž 3.8.
Hence, it follows from Ž3.5., Ž3.6., and Ž3.8. that
Ž1 y x.
2
f0 Ž x.
1yt )
d
ž /
y
c
2
`
Ý ns0
c Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q Ž n q p q 1 . t n d Ž p, n q 2 . n!
ab Ž a q 1 . Ž b q 1 . p Ž p q 1.
Ž 1 y t . F Ž a q 2, b q 2; p q 2; t .
227
PRECISE ESTIMATES FOR DIFFERENCES
s
ž /
ns0
`
Ž a, n q 1 . Ž b, n q 1 . Ž n q ab q p q 1 . t n Ž p, n q 2 . n!
Ý ns0
s
`
2
d
c Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q Ž n q p q 1 . t n d Ž p, n q 2 . n!
Ý
c
y
`
2
d
ž /
Ž a, n q 1 . Ž b, n q 1 . Ž n q ab q p q 1 . Ž p, n q 2 . n!
Ý
c
ns0
¡ c =~y ž / ¢ d s
ž /
ns1
¡ c =~y ž / ¢ d s
d
ž / c
ns1
½
n
c
Ž n q p. d Ž a, n . Ž b, n . Ž n q ab q p .
q
¦¥ §t
ny 1
Ž a, n . Ž b, n . Ž n q ab q p . Ž p, n q 1 . Ž n y 1 . !
Ý
= QŽ n.
Ž u, n . Ž ¨ , n . u¨ q
2
`
2
¦¥ §t
Ž a, n . Ž b, n . Ž n q ab q p . Ž p, n q 1 . Ž n y 1 . !
Ý
c
c
Ž n q p q 1. d Ž a, n q 1 . Ž b, n q 1 . Ž n q ab q p q 1 .
q
`
2
d
Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q
2
G Ž a. G Ž b . G Ž u. G Ž ¨ .
y
2
c
ž /5 d
t ny 1 ,
Ž 3.9.
where QŽ n. is as in Lemma 2.9. Hence, it follows from Lemma 2.9 and Ž3.9. that
Ž 1 y x . 2y ) s
Ž drc .
f0 Ž x.
d G Ž a. G Ž b . c G Ž u. G Ž ¨ . 1
d
B Ž u, ¨ .
c
=
`
Ý ns1
y
c d
`
Ý ns1
Ž a, n . Ž b, n . Ž n q ab q p . ny 1 t Ž p, n q 1 . Ž n y 1 . !
B Ž a, b . y B Ž u, ¨ .
Ž a, n . Ž b, n . Ž n q ab q p . ny 1 t . Ž p, n q 1 . Ž n y 1 . !
Ž 3.10.
By Lemma 2.11, as a function of d , Ž drc. B Ž a, b . y B Ž u, ¨ . is strictly decreasing on Ž0, a.. Since 0 - d F d 1 , we have d d B Ž a, b . y B Ž u, ¨ . G B Ž a, b . y B Ž a y d 1 , b q d 1 . s 0. c c
228
ANDERSON, QIU, AND VUORINEN
Hence, it follows from Ž3.10. that f 0 Ž x . ) 0 for all x g Ž0, 1.. This shows that f 9Ž x . is strictly increasing on Ž0, 1., and so is h1Ž x . by Ž3.2., Ž3.3., and Lemma 2.19. The asserted monotoneity of h now follows. By l’Hopital’s Rule, Ž3.2., and Ž3.3. we get ˆ u¨ d
h Ž 1y . s h1 Ž 0q . s f 9 Ž 0 . s s
d cp
ab 1 y
½ ž
c d
cp
y
ab p
y d 2 q Ž b y a. d
/
5
s A1 Ž d . . Ž 3.11.
By Lemma 2.15, for 0 - d F d 1 , we have A1Ž d . G A1Ž d 1 . ) A1Ž d 2 . s 0. On the other hand, since F Ž a, b; a q b; x . s
`
1
Ž a, n . Ž b, n .
Ý B Ž a, b . ns0
Ž n! .
2
2C Ž n q 1 . y C Ž a q n .
yC Ž b q n . y log Ž 1 y x . Ž 1 y x .
n
Ž 3.12.
by w1, 15.3.10x, and, since C Ž1. s yg w1, 6.3.2x, we have h Ž 0q . s h1 Ž 1y . s f Ž 1y . s limy xª1
1
½
B Ž u, ¨ . y
s limy xª1
½
R Ž u, ¨ . y log Ž 1 y t .
1
R Ž a, b . y log Ž 1 y x .
B Ž a, b .
R Ž u, ¨ . B Ž u, ¨ . =
y d c
R Ž a, b . B Ž a, b .
q
5
1 B Ž u, ¨ . B Ž a, b .
B Ž a, b . y B Ž u, ¨ . log
1 1yx
s A2 Ž d . .
5 Ž 3.13.
Here, in the last equality, we have used Lemma 2.11. Formula Ž1.11. and the first inequality in Ž1.12. are clear, while the other inequalities in Ž1.12. and the equality case follow from Lemma 2.13Ž2.. For part Ž2., we observe that, for d 1 - d - d 2 , Ž3.11. and the first four equalities in Ž3.13. still hold. Since A1Ž d . is strictly decreasing in d , h Ž 1y . s A1 Ž d . ) A1 Ž d 2 . s 0,
d1 - d - d2 .
229
PRECISE ESTIMATES FOR DIFFERENCES
By Lemma 2.11, d c
B Ž a, b . y B Ž u, ¨ . -
d c
B Ž a, b . y B Ž a y d 1 , b q d 1 . s 0
for all d 1 - d - d 2 , so that from Ž3.13. we get h Ž 0q . s y`,
d1 - d - d2 .
Hence, part Ž2. follows. 3.14. COROLLARY. Under the hypotheses of Theorem 1.7Ž1., the function f Ž x . ' F Ž a y d , b q d ; a q b; 1 y Ž 1 y x .
drc
. y F Ž a, b; a q b; x .
is strictly increasing and con¨ ex from Ž0, 1. onto Ž0, A 2 ., where A 2 s A 2 Ž d . is as in Theorem 1.7. Proof. In the proof of Theorem 1.7Ž1. we showed that f 9Ž x . is positive and strictly increasing on Ž0, 1. and that f Ž0. s f Ž1y . y A 2 s 0. 3.15. Proof of Theorem 1.13. As in the proof of Theorem 1.7, let u s a y d , ¨ s b q d s p y u, t s 1 y Ž1 y x . d r c, and let h1 Ž x . s
1 c
hŽ Ž1 y x .
1rc
.s
F Ž u, ¨ ; p; t . y F Ž a, b; p; x . log Ž 1r Ž 1 y x . .
.
In order to prove the first statement, we need only show that h1 is strictly decreasing from Ž0, 1. onto Ž A 3rc, A 4rc . if d 2 F d - a. For this, in the notation of Lemma 2.19 let f Ž x . s F Ž u, ¨ ; p; t . y F Ž a, b; p; x . ,
g Ž x . s log
1 1yx
.
Then f Ž0. s g Ž0. s 0 and fXŽ x. X
g Ž x.
s h2 Ž x . ' Ž 1 y x .
u¨ d cp
Ž1 y x.
Ž drc .y1
y s
u¨ d cp y
ab p
F Ž u q 1, ¨ q 1; p q 1; t .
F Ž a q 1, b q 1; p q 1; x .
Ž 1 y t . F Ž u q 1, ¨ q 1; p q 1; t .
ab p
Ž 1 y x . F Ž a q 1, b q 1; p q 1; x . .
Ž 3.16.
230
ANDERSON, QIU, AND VUORINEN
Since h1Ž x . s f Ž x .rg Ž x ., the desired monotoneity will follow from Lemma 2.19 if we prove that h 2 Ž x . is decreasing. As in the proof of Theorem 1.7 we shall do this by expressing the hypergeometric functions in Ž3.16. in terms of power series and comparing coefficients. By Ž3.16. and l’Hopital’s Rule, we get ˆ h1 Ž 0q . s h 2 Ž 0 . s
u¨ d cp
ab
y
p
s
A4 c
,
which is nonpositive by the definition of d 2 and the monotoneity of A 4 Ž d .. On the other hand, Ž3.12. yields h1 Ž 1y . s lim
xª1
1 log Ž 1r Ž 1 y x . .
½
1 B Ž u, ¨ . 1
y s
d
y
cB Ž u, ¨ .
1 B Ž a, b .
s
B Ž a, b . A3 c
d
R Ž u, ¨ . y
c
log Ž 1 y x .
R Ž a, b . y log Ž 1 y x .
5
.
Next, differentiation Žsee Ž3.16.. gives
Ž 1 y x . hX2 Ž x . s h 3 Ž x . ' y y
pc 2
Ž 1 y t . F Ž u q 1, ¨ q 1; p q 1; t .
Ž u q 1. Ž ¨ q 1. pq1 q
y
u¨ d 2
Ž a q 1. Ž b q 1. pq1
ab p
2 Ž 1 y t . F Ž u q 2, ¨ q 2; p q 2; t .
Ž 1 y x . F Ž a q 1, b q 1; p q 1; x .
2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . . Ž 3.17.
Since p q 1 s Ž u q 1. q Ž ¨ q 1. y 1, and since u¨ s y d 2 q Ž b y a . d y ab 1 y
ž
c d
F y d 22 q Ž b y a . d 2 y ab 1 y
ž
s
abc d
F 1,
q
/
c d
/
abc
q
d abc d
Ž 3.18.
PRECISE ESTIMATES FOR DIFFERENCES
231
it follows from Lemma 2.14 that the expression in the first set of brackets in Ž3.17. is strictly increasing in t on Ž0, 1.. Hence, since x - t, u¨ d 2
h3 Ž x . - y
Ž 1 y x . F Ž u q 1, ¨ q 1; p q 1; x .
pc 2
Ž u q 1. Ž ¨ q 1.
y q
ab
pq1
2 Ž 1 y x . F Ž u q 2, ¨ q 2; p q 2; x .
Ž 1 y x . F Ž a q 1, b q 1; p q 1; x .
p
Ž a q 1. Ž b q 1.
y
pq1
2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . ,
so that Ž3.17. and Ž2.6. yield
hX2 Ž x . - y
u¨ d 2 pc 2
F Ž u q 1, ¨ q 1; p q 1; x . y
ab
q
p
d2 c
2
pq1
`
Ý ns0
Ž a q 1. Ž b q 1. pq1
`
Ý ns0
`
Ý ns0
Ž 1 y x . F Ž a q 2, b q 2; p q 2; x .
Ž u, n q 1 . Ž ¨ , n q 1 . n x Ž p, n q 1 . n!
yŽ 1 y x . q
Ž 1 y x . F Ž u q 2, ¨ q 2; p q 2; x .
F Ž a q 1, b q 1; p q 1; x . y
sy
Ž u q 1. Ž ¨ q 1.
Ž u, n q 2 . Ž ¨ , n q 2 . n x Ž p, n q 2 . n!
Ž a, n q 1 . Ž b, n q 1 . n x Ž p, n q 1 . n!
yŽ 1 y x .
`
Ý ns0
Ž a, n q 2 . Ž b, n q 2 . n x Ž p, n q 2 . n!
232
ANDERSON, QIU, AND VUORINEN
sy
`
2
d
ž /
Ý
c
ns0
Ž u, n q 1 . Ž ¨ , n q 1 . Ž p, n q 2 . n!
= Ž n q 1. Ž p q n q 1. y Ž u q n q 1. Ž ¨ q n q 1. x n `
q
Ž a, n q 1 . Ž b, n q 1 . Ž n q 1. Ž p q n q 1. Ž p, n q 2 . n!
Ý ns0
y Ž a q n q 1. Ž b q n q 1. x n s
d
ž /
Ý
c
y
`
2
u¨ Ž u, n q 1 . Ž ¨ , n q 1 .
Ž p, n q 2 . n!
ns0
`
ab Ž a, n q 1 . Ž b, n q 1 .
Ý
Ž p, n q 2 . n!
ns0
s s
`
Ž a, n . Ž b, n .
u¨
Ý ns1 Ž p, n q 1 . Ž n y 1 . ! `
xn
xn 2
d
ž / ŽŽ c
u, n . Ž ¨ , n . a, n . Ž b, n .
y ab x ny 1
Ž a, n . Ž b, n .
Ý ns1 Ž p, n q 1 . Ž n y 1 . !
= u¨
2
d
ž /
G Ž n q u . G Ž n q ¨ . G Ž a. G Ž b . G Ž n q a. G Ž n q b . G Ž u . G Ž ¨ .
c
y ab x ny 1 .
Thus it follows from Lemma 2.10 and Ž2.6. that hX2
Ž x. -
d
1
2
ž /
ab
c
2
Ž u¨ . y
abc
`
2
ž /
Ž a, n . Ž b, n .
x ny1 , Ý ns1 Ž p, n q 1 . Ž n y 1 . !
d
which is nonpositive by Ž3.18.. Hence, h 2 is strictly decreasing on Ž0, 1.. The monotoneity of h1 now follows from Lemma 2.19 and Ž3.16.. The first inequality in Ž1.17. follows from Lemma 2.13Ž2., while Ž1.16., the other parts of Ž1.17., and the equality case are clear. 3.19. Proof of Corollary 1.19. For part Ž1., we observe that if a s b s 1r2 in Theorem 1.7, then d 1 satisfies cB
ž
1 2
y d1 ,
1 2
q d 1 s dB
/
1 1 , 2 2
ž /
or, equivalently, cG
ž
1 2
y d1 G
/ž
1 2
q d 1 s dG
/
1
ž / 2
2
s p d,
Ž 3.20.
233
PRECISE ESTIMATES FOR DIFFERENCES
so that, by Ž2.7.,
d1 s
1
p
arccos
c
ž / d
,
since Ž1r2. q d 1 s 1 y wŽ1r2. y d 1 x. Clearly, d 2 s Next, it follows from Ž2.5., Ž3.20., and Ž1.10. that A2 Ž d 1 . s
c
1
p d
R
ž
1 2
y d1 ,
1 2
1 2
'1 y Ž crd . .
q d 1 y log 16 ,
/
while Ž1.20. holds by Ž1.9.. Part Ž1. now follows from Theorem 1.7Ž1.. By Theorem 1.7Ž2., part Ž3. is clear. For part Ž2., by Theorem 1.13 we need only verify that ab F drc for a s b s 1r2. But this is clear. 3.21. Remark. Ž1. In Theorems 1.7 and 1.13, if a q b s p s 1, then, by Ž2.7.,
¡B Ž a y d , b q d . s G Ž a y d . G Ž 1 y Ž a y d . . s
~
¢
B Ž a, b . s G Ž a . G Ž 1 y a . s
p sin ap
p sin Ž a y d . p
,
,
and hence d 1 has the explicit expression
d1 s
1
p
½
ap y arcsin
ž
c d
sin ap
/5
say
1 2
q
1
p
arccos
ž
c d
sin ap .
/
In particular, if a s 1r2, then d 1 s Ž1rp .arccosŽ crd .. Ž2. Our computational work supports the validity of the following conjecture: The condition ‘‘abc F d’’ in Theorem 1.13 is not necessary.
ACKNOWLEDGMENTS This research was completed during the second author’s visit to Michigan State University, with grants from its Department of Mathematics and from the Academy of Finland. The authors are grateful to the referee for helpful suggestions for the improvement of the presentation.
234
ANDERSON, QIU, AND VUORINEN
REFERENCES 1. M. Abramowitz and I. A. Stegun ŽEds.., ‘‘Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,’’ Dover, New York, 1965. 2. L. V. Ahlfors, ‘‘Complex Analysis,’’ 2nd ed., McGraw]Hill, New York, 1966. 3. G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 Ž1995., 1713]1723. 4. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Hypergeometric functions and elliptic integrals, in ‘‘Current Topics in Analytic Function Theory’’ ŽH. M. Srivastava and S. Owa, Eds.., pp. 48]85, World Scientific, SingaporerLondon, 1992. 5. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for quasiconformal mappings in space, Pacific J. Math. 160 Ž1993., 1]18. 6. R. Askey, Ramanujan and hypergeometric and basic hypergeometric series, in ‘‘Ramanujan Internat. Symposium on Analysis, December 26]28, 1987’’ ŽN. K. Thakare, Ed.., Pune, India. 7. B. C. Berndt, ‘‘Ramanujan’s Notebooks,’’ Part II, Springer-Verlag, BerlinrHeidelbergrNew York, 1989. 8. B. C. Berndt, S. Bhargava, and F. G. Garvan, Ramanujan’s theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347 Ž1995., 4163]4244. 9. J. M. Borwein and P. B. Borwein, Inequalities for compound mean iterations with logarithmic asymptotes, J. Math. Anal. Appl. 177 Ž1993., 572]582. 10. S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 Ž1997., 278]301. 11. E. D. Rainville, ‘‘Special Functions,’’ Macmillan, New York, 1960. 12. E. T. Whittaker and G. N. Watson, ‘‘A Course of Modern Analysis,’’ Cambridge Univ. Press, Cambridge, 1958.
215, 212]234 Ž1997.
AY975641
Precise Estimates for Differences of the Gaussian Hypergeometric Function G. D. Anderson* Department of Mathematics, Michigan State Uni¨ ersity, East Lansing, Michigan 48824
S.-L. Qiu Hangzhou Institute of Electronics Engineering (HIEE), Hangzhou, 310037, People’s Republic of China
and M. Vuorinen† Department of Mathematics, Uni¨ ersity of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014, Helsinki, Finland Submitted by Bruce C. Berndt Received July 15, 1996
By showing certain combinations of the Gaussian hypergeometric functions F Ž a, b; a q b; 1 y x c . and F Ž a y d , b q d ; a q b; 1 y x d . to be monotone on Ž0, 1. for given a, b, c, d g Ž0, `., a F b, and c - d, the authors study the problem of comparing these two functions. They find sup d g Ž0, a. : F Ž a, b; a q b; 1 y x c . F Ž a y d , b q d ; a q b; 1 y x d . for x g Ž0, 1.4, thus solving an open problem and a recent conjecture. Q 1997 Academic Press
1. INTRODUCTION In this paper we consider the Gaussian hypergeometric function F Ž a, b; c; x . s2 F1 Ž a, b; c; x . '
`
Ý ns0
* E-mail address: [email protected]. † E-mail address: [email protected]. 212 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
Ž a, n . Ž b, n . x n n! Ž c, n .
Ž 1.1.
213
PRECISE ESTIMATES FOR DIFFERENCES
for x g Žy1, 1., where Ž a, n. denotes the shifted factorial function Ž a, n. ' aŽ a q 1. ??? Ž a q n y 1., n s 1, 2, . . . , and Ž a, 0. s 1 for a / 0. In studying connections between the arithmetic-geometric mean and other mean values, J. M. Borwein and P. B. Borwein w9, Ž2.9.x showed that F
ž
1 1 1 1 , ; 1; 1 y x 2 - F y d , q d ; 1; 1 y x 3 2 2 2 2
/ ž
/
Ž 1.2.
for d s 1r6 and all x g Ž0, 1.. In w4, p. 79x it was conjectured that Ž1.2. holds for all d g Ž0, 1r6x and all x g Ž0, 1.. In w3, Theorem 4.3x it was proved that F
ž
1 1 1 1 , ; 1; 1 y x c - F y d 0 , q d 0 ; 1; 1 y x d 2 2 2 2
/ ž ž -F
1 2
yd ,
1 2
/
qd ; 1; 1 y x d - F
/ ž
1 1 , ; 1; 1 y x d 2 2 Ž 1.3.
/
for all d g Ž0, d 0 ., x g Ž0, 1., and c, d g Ž0, `. with 0 - 4 c - p d - `, where d 0 s ŽŽ dp y 4 c .rŽ4p d ..1r2 . It was conjectured in w3, Conjecture 4.10Ž1.x that for c s 2, d s 3 the best value of d 0 for which Ž1.3. is valid is
d0 s
1 2p
ž
p y 2 arcsin
2 3
/
s
1
p
arccos
2 3
f 0.268.
It is natural to ask what is the best value of d 0 s d 0 Ž c, d ., for which Ž1.3. holds for all x g Ž0, 1. and 0 - c - d - `. For the more general case, the following open problem is of interest. 1.4. Open Problem w4, p. 79, Open Problem Ž2.x. It is true, for small values of d , say 0 - d - min a, b4 , that F Ž a, b; a q b; 1 y x c . - F Ž a y d , b q d ; a q b; 1 y x d .
Ž 1.5.
for x g Ž0, 1., a, b, c, d g Ž0, `. with 0 - c - d - `? Considering the special case mentioned above, we can naturally raise the following two questions. 1.6. QUESTIONS. Ž1. For gi¨ en a, b, c, d g Ž0, `., c - d, what is the best ¨ alue of d 0 such that Ž1.5. is ¨ alid for all d g Ž0, d 0 x and all x g Ž0, 1.? Ž2. What can we conclude about the comparison between two functions in Ž1.5. for d 0 - d - min a, b4 if we can find d 0 in Question Ž1.?
214
ANDERSON, QIU, AND VUORINEN
In the present paper, we study Questions 1.6Ž1. and 1.6Ž2.. We shall not only find the full solution to Question 1.6Ž1., but also obtain error estimates for the difference of the two functions appearing in Ž1.5. whenever they are directly comparable for all x g Ž0, 1. Žsee Theorem 1.7 below.. In particular, our results imply the truth of w3, Conjecture 4.10Ž1.x for more general c and d Žsee Corollary 1.19. and give the answer to Open Problem 1.4. We shall also answer Question 1.6Ž2. in the case abc F d Žsee Theorem 1.13 below.. We now state our main results. 1.7. THEOREM. Let a, b, c, d g Ž0, `., a F b, p s a q b, c - d, and let d 1 be the unique solution for d of the equation cB Ž a y d , b q d . s dB Ž a, b . ,
d ) 0,
Ž 1.8.
where B Ž x, y . is the beta function, and let d 2 denote the unique positi¨ e root of the quadratic equation
d 2 q Ž b y a . d y ab 1 y
ž
c d
/
s 0,
that is,
d2 s
1
½
2
2 Ž b y a. q 4 ab 1 y
ž
c d
1r2
/
5
y Ž b y a. .
Then we ha¨ e: Ž1.
For 0 - d F d 1 , the function
hŽ x . '
1 1 y xc
F Ž a y d , b q d ; p; 1 y x d . y F Ž a, b; p; 1 y x c .
is strictly decreasing from Ž0, 1. onto Ž A1 , A 2 ., where A1 s A1 Ž d . '
A2
d cp
ab 1 y
½ ž
c d
y d 2 q Ž b y a. d
¡`, R Ž a, b . ¢B Ž a y d , b q d . y B Ž a, b . ,
s A Ž d . '~ R Ž a y d , b q d . 2
/
5
) 0,
Ž 1.9.
if 0 - d - d 1 , if d s d 1 ,
Ž 1.10.
215
PRECISE ESTIMATES FOR DIFFERENCES
while RŽ a, b . is defined by Ž2.4.. In particular, for all x g Ž0, 1. and 0 d F d 1, F Ž a, b; p; 1 y x c . q Ž 1 y x c . A1 - F Ž a y d , b q d ; p; 1 y x d . - F Ž a, b; p; 1 y x c . q Ž 1 y x c . A 2 ,
Ž 1.11. F Ž a, b; p; 1 y x c . q Ž 1 y x c . A1 - F Ž a y d 1 , b q d 1 ; p; 1 y x d . F F Ž a y d , b q d ; p; 1 y x d . - F Ž a, b; p; 1 y x d . , Ž 1.12. with equality in the second inequality in Ž1.12. if and only if d s d 1. Ž2. If d 1 - d - d 2 then, as functions of x, F Ž a y d , b q d ; p; 1 y x d . and F Ž a, b; p; 1 y x c . are not directly comparable on Ž0, 1., that is, neither F Ž a, b; p; 1 y x c . F F Ž a y d , b q d ; p; 1 y x d . nor its re¨ ersed inequality holds for all x g Ž0, 1.. In our next main result we study the case d 2 F d - min a, b4 . 1.13. THEOREM. Let a, b, c, d, p, d 1 , and d 2 be as in Theorem 1.7, and abc F d. Then, for d 2 F d - a, the function hŽ x . '
1 log Ž 1rx .
F Ž a y d , b q d ; p; 1 y x d . y F Ž a, b; p; 1 y x c .
is strictly increasing from Ž0, 1. onto Ž A 3 , A 4 ., where A3 s A3 Ž d . '
d BŽ a y d , b q d .
c
y
B Ž a, b .
Ž 1.14.
and A4 s A4 Ž d . ' y
d p
d 2 q Ž b y a . d y ab 1 y
ž
c d
/
F 0. Ž 1.15.
In particular, for all x g Ž0, 1. and d g w d 2 , a., F Ž a, b; p; 1 y x c . q A 3 Ž d . log
1 x
- F Ž a y d , b q d ; p; 1 y x d . - F Ž a, b; p; 1 y x c . q A 4 Ž d . log
1 x
- F Ž a, b; p; 1 y x c . , Ž 1.16.
216
ANDERSON, QIU, AND VUORINEN
F Ž a y d , b q d ; p; 1 y x d . F F Ž a y d 2 , b q d 2 ; p; 1 y x d . - F Ž a, b; p; 1 y x c . q A 4 Ž d 2 . log
1 x
s F Ž a, b; p; 1 y x c . , Ž 1.17.
with equality in the first inequality in Ž1.17. if and only if d s d 2 . 1.18. Remark. By symmetry of the function F Ž a, b; a q b; x . with respect to the parameters a and b, without loss of generality we may assume that a F b for convenience, as we did in Theorems 1.7 and 1.13. 1.19. COROLLARY. Ž1. For Ž1rp .arccosŽ crd ., the function
hŽ x . '
1 1yx
F
c
ž
1 2
y d,
1 2
0 - c - d - ` and 0 - d F d 1 '
q d ; 1; 1 y x d y F
/ ž
1 1 , ; 1; 1 y x c 2 2
/
is strictly decreasing from Ž0, 1. onto Ž A1 , A 2 ., where
A1 s A1 Ž d . s
A2
d 1 c 4
ž
1y
c d
y d 2 ) 0,
/
¡`, c ¢p ž d R y log 16 / ,
if 0 - d - d 1 ,
s A Ž d . s~ 1 2
Ž 1.20.
if d s d 1 ,
Ž 1.21.
and R s RŽ 12 y d , 12 q d ., while RŽ a, b . is defined by Ž2.4.. In particular, for all x g Ž0, 1., 0 - c - d - `, and 0 - d F d 1 ,
F
ž
1 1 , ; 1; 1 y x c q A1 Ž d . Ž 1 y x c . 2 2
/
-F -F
ž ž
1 2
y d1 ,
1 2
q d 1 ; 1; 1 y x d
/
1 1 1 c , ; 1; 1 y x c q R y log 16 Ž 1 y x c . 2 2 p d
/ ž
/
Ž 1.22.
217
PRECISE ESTIMATES FOR DIFFERENCES
and F
ž
1 1 , ; 1; 1 y x c q A1 Ž d 1 . Ž 1 y x c . 2 2
/
-F -F
ž ž
1
y d1 ,
2
1 2
q d 1 ; 1; 1 y x d F F
/ ž
1 2
yd,
1 2
q d ; 1; 1 y x d
1 1 , ; 1; 1 y x d , 2 2
/
/
Ž 1.23.
with equality if and only if d s d 1. Ž2. For 0 - c - d - ` and 12 1 y Ž crd . ' d 2 F d - 12 , the function
'
HŽ x. '
1
F
log Ž 1rx .
ž
1 2
yd,
1 2
q d ; 1; 1 y x d y F
/ ž
1 1 , ; 1; 1 y x c 2 2
/
is strictly increasing from Ž0, 1. onto Ž A 3 , A 4 ., where d
A3 s A3 Ž d . s
pž
A4 s A4 Ž d . s d
cos dp y
1 4
ž
1y
c d
/
c d
/
,
Ž 1.24.
y d 2 F 0.
Ž 1.25.
In particular, for all x g Ž0, 1., 0 - c - d - `, and d 2 F d F 12 , F
ž
1 1 1 1 1 , ; 1; 1 y x c q A 3 Ž d . log - F y d , q d ; 1; 1 y x d 2 2 x 2 2
/
-F
F
ž
1 2
yd, FF -F
ž ž
1 2 1 2
q d ; 1; 1 y x d y d2 ,
1 2
ž ž
/
1 1 1 , ; 1; 1 y x c q A 4 Ž d . log , 2 2 x Ž 1.26.
/
/
q d 2 ; 1; 1 y x d
/
1 1 1 1 1 , ; 1; 1 y x c q A 4 Ž d 2 . log s F , ; 1; 1 y x c . 2 2 x 2 2 Ž 1.27.
/
ž
Equality holds in the first inequality in Ž1.27. if and only if d s d 2 .
/
218
ANDERSON, QIU, AND VUORINEN
0 - c - d - `, if Ž1r p .arccosŽ crd . ' d 1 - d - d 2 ' then, as functions of x, F Ž 12 , 12 ; 1; 1 y x c . and F Ž 12 y d , d. q d ; 1; 1 y x are not directly comparable on Ž0, 1., that is, neither Ž3 .
1 2 1 2
For
'1 y Ž crd . , F
ž
1 1 1 1 , ; 1; 1 y x c - F y d , q d ; 1; 1 y x d 2 2 2 2
/ ž
/
nor its re¨ ersed inequality holds for all x g Ž0, 1.. Proofs of Theorems 1.7 and 1.13 and Corollary 1.19 will be supplied in Section 3. These theorems imply that, for a, b, c, d g Ž0, `., c - d, sup d : Ž 1.5. holds for all x g Ž 0, 1 . 4 s d 1 , where d 1 is as in Theorem 1.7. Furthermore, they give the correct asymptotic behavior of, and provide error estimates for, the difference of the two functions in Ž1.5. whenever they are directly comparable for all x g Ž0, 1..
2. PRELIMINARY RESULTS Before we prove our main results stated in Section 1, we need to establish several technical lemmas. But first, let us recall some known results for F Ž a, b; c; x . and for the gamma function. One of the tools we shall need in our work is the following asymptotic limit w12, Example 18, p. 299; 10, Theorem 1.10x, valid for c - a q b, limy Ž 1 y x .
aq byc
xª1
F Ž a, b; c; x . s
GŽ c. GŽ a q b y c. G Ž a. G Ž b .
,
Ž 2.1.
where G denotes the classical gamma function. As illustrated by Ž2.1., properties of the hypergeometric function are closely associated with the gamma function, its logarithmic derivative C Ž x ., and the beta function B Ž x, y .. For positive x and y these functions are defined by GŽ x. s
`
yt xy1
H0 e
t
dt, C Ž x . s
G9 Ž x . GŽ x.
, B Ž x, y . s
GŽ x. GŽ y. GŽ x q y.
Ž 2.2.
Žcf. w12x.. We shall need the Euler]Mascheroni constant g defined by n
g s lim
nª`
ž
Ý ks1
1 k
/
y log n s 0.57721 . . . ,
Ž 2.3.
219
PRECISE ESTIMATES FOR DIFFERENCES
and the function R Ž a, b . ' y2g y C Ž a . y C Ž b . .
Ž 2.4.
By w1, 6.3.3x, we have R
1 1 1 , s y2 g q C 2 2 2
ž
ž /
ž //
s log 16.
Ž 2.5.
The gamma function satisfies the difference equation w12, p. 237x G Ž x q 1 . s xG Ž x .
Ž 2.6.
if x is not 0 or a negative integer. It has the so-called reflection property w12, p. 239x GŽ x. GŽ1 y x. s
p sin p x
Ž 2.7.
if x is not an integer. We shall also need Stirling’s well-known asymptotic formula lim e x x 1r2yx G Ž x . s '2p
xª`
Ž 2.8.
w12, p. 251x. For other properties of these functions the reader is referred to w1, 6]8x. 2.9. LEMMA. a, the sequence
For a, b, c, d g Ž0, `., a F b, and c - d, and for 0 - d -
QŽ n. '
G Ž n qa yd . G Ž n qb qd . Ž a yd . Ž b qd . q Ž crd . Ž n qa qb . G Ž n qa . G Ž n qb . Ž n qabq a q b .
,
for n g N, is strictly decreasing, with lim n ª` QŽ n. s crd. Here, and in the sequel, N denotes the set of natural numbers. Proof. Let p s a q b, u s a y d , ¨ s b q d , and let Q1 Ž n . s Ž n2 q pn q u¨ . u¨ q
c d
Ž n q p q 1 . Ž n q ab q p .
y Ž n2 q pn q ab . u¨ q
c d
Ž n q p . Ž n q ab q p q 1 .
220
ANDERSON, QIU, AND VUORINEN
for n g N. Since, by Ž2.6., Q Ž n q 1. QŽ n.
s
Ž n2 q pn q u¨ .
u¨ q
Ž n2 q pn q ab .
u¨ q
c d c d
Ž n q p q 1 . Ž n q ab q p . Ž n q p . Ž n q ab q p q 1 .
for n g N, QŽ n. is strictly decreasing if and only if Q1Ž n. - 0 for all n g N. By elementary computation we obtain Q1 Ž n . s u¨
ž
c d
c
½
y 1 n2 q Ž u¨ y ab . u¨ q
/
Ž ab q p q 1 .
d
qpu¨ q Ž u¨ y ab . Ž ab q p . u¨ q
ž
cp d
/
q
cp d
ž
c d
y1
q abu¨
ž
/5 c d
n y1 ,
/
which is negative for all n g N since c - d and u¨ s ab y d Ž d q b y a. F ab. From Ž2.8., it follows that lim Q Ž n . s
nª`
s
c d
G Ž n q a. G Ž n q b .
nª`
c d
G Ž n q u. G Ž n q ¨ .
lim
lim
nª`
ž
= 1q
ž
s 2.10. LEMMA.
c d
nqu nqa
ay 1r2
/
ž
u¨ y ab n2 q Cn q ab
nq¨ nqb
by1r2
/
ž
nq¨ nqu
d
/
n
/
.
For a, b, u, ¨ g Ž0, `. with a q b s u q ¨ , and for n g N,
let QŽ n. s
G Ž n q u. G Ž n q ¨ . G Ž n q a. G Ž n q b .
.
Ž1. If u¨ s ab, then QŽ n. s 1 for all n g N. Ž2. If u¨ - ab Ž u¨ ) ab., then QŽ n. is strictly decreasing Ž increasing ., and lim n ª` QŽ n. s 1.
221
PRECISE ESTIMATES FOR DIFFERENCES
Proof. By Ž2.6., we have Q Ž n q 1. QŽ n.
s1q
u¨ y ab n q Ž a q b . n q ab 2
,
which clearly implies part Ž1. and the monotoneity of QŽ n. in part Ž2.. It follows from Ž2.8. that lim Q Ž n .
nª`
s lim
nª`
Ž n q u.
nq uy1r2
Žnq¨.
nq¨ y1 r2
Ž n q a.
nq ay1r2
Ž n q b.
nqby1r2
s lim 1 q nª`
n
u¨ y ab n2 q Ž a q b . n q ab
ž
nqu nqa
a
/ž
nqu nqb
b
/ž
nq¨ nqu
¨
/
s 1. 2.11. LEMMA.
For a, b g Ž0, `. with a F b, the functions
f 1Ž x . ' G Ž a y x . G Ž b q x . ,
f 2 Ž x . ' B Ž a y x, b q x . ,
and f 3 Ž x . ' R Ž a y x, b q x . are all strictly increasing and con¨ ex on Ž0, a., where R is as in Ž2.4.. Proof. Since C Ž x . s yg y
1 x
q
`
x
Ý ns1 n Ž n q x .
,
C9 Ž x . s
`
Ý ns0
1
Ž n q x.
2
Ž 2.12.
Žcf. w2, p. 198x. we see that C is strictly increasing on Ž0, `., while C9 is strictly decreasing there. Logarithmic differentiation gives f 1X Ž x . s f 1 Ž x . C Ž b q x . y C Ž a y x . , which is positive and, further, a product of two positive and strictly increasing functions on Ž0, a.. Hence the result for f 1 follows. Since f 2 Ž x . s f 1Ž x .rG Ž a q b ., the assertion about f 2 follows. By differentiation, we obtain f 3X Ž x . s C9 Ž a y x . y C9 Ž b q x . , which is positive and strictly increasing on Ž0, a..
222
ANDERSON, QIU, AND VUORINEN
2.13. LEMMA. Ž1. For n g N and c g Ž0, `., as a function of a, f nŽ a. ' C Ž a q n. y C Ž c y a q n. y C Ž a. q C Ž c y a. is strictly decreasing on Ž0, c . with f nŽ cr2. s 0. Ž2. For c g Ž0, `. and x g Ž0, 1., as a function of a, hŽ a. ' F Ž a, c y a; c; x . is strictly increasing on Ž0, cr2x, and decreasing on w cr2, c .. Proof. Ž1. Since C9Ž x . is strictly decreasing on Ž0, `. by Ž2.12., we get f nX Ž a . s C9 Ž a q n . y C9 Ž a . q C9 Ž c y a q n . y C9 Ž c y a . - 0, and hence the monotoneity of f n follows. Clearly, f nŽ cr2. s 0. Ž2. Since g n Ž a . ' Ž a, n . Ž c y a, n . s
G Ž a q n. G Ž c y a q n. G Ž a. G Ž c y a.
,
logarithmic differentiation gives g Xn Ž a . s g n Ž a . f n Ž a . ,
n g N,
so that, by Ž1.1., h9 Ž a . s
`
g n Ž a.
Ý Ž c, n . n! f n Ž a. x n . ns0
Hence part Ž2. follows from part Ž1.. 2.14. LEMMA.
For a, b g Ž1, `. with Ž a y 1.Ž b y 1. F 1, the function
h Ž x . ' Ž 1 y x . F Ž a, b; a q b y 1; x . y
ab aqby1
2 Ž 1 y x . F Ž a q 1; b q 1; a q b; x .
is strictly increasing and con¨ ex from Ž0, 1. onto Ž1 y abrŽ a q b y 1., 0.. Proof. Clearly hŽ0. s 1 y abrŽ a q b y 1., while Ž2.1. and Ž2.6. yield h Ž 1y . s
G Ž a q b y 1. G Ž 1. G Ž a. G Ž b .
y
ab
G Ž a q b . G Ž 2.
a q b y 1 G Ž a q 1. G Ž b q 1.
s 0.
PRECISE ESTIMATES FOR DIFFERENCES
223
Using the series expansion of F Ž a, b; c; x ., we get
hŽ x . s Ž 1 y x .
`
Ž a, n . Ž b, n .
Ý Ž a q b y 1, n . n! x n ns0 yŽ 1 y x .
s Ž1 y x.
`
`
Ž a, n q 1 . Ž b, n q 1 .
Ž a, n . Ž b, n .
xn Ý ns0 Ž a q b y 1, n . n! y
q
`
xn Ý ns0 Ž a q b y 1, n q 1 . n!
`
Ž a, n q 1 . Ž b, n q 1 .
xn Ý ns0 Ž a q b y 1, n q 1 . n!
n Ž a, n . Ž b, n .
xn Ý ns0 Ž a q b y 1, n . n!
s Ž 1 y x . Ž a q b y 1 y ab . s Ž x y 1.
`
Ý ns0
s Ž x y 1.
`
Ý ns0
s
`
`
Ý ns0
s
`
Ž a y 1, n q 1 . Ž b y 1, n q 1 . n x Ž a q b y 1, n q 1 . n!
Ž a q b y 1, n . n!
ns0
Ž a, n . Ž b, n .
Ž a y 1 . Ž b y 1 . Ž a, n . Ž b, n . n x Ž a q b y 1, n q 1 . n!
n Ž a y 1, n . Ž b y 1, n .
Ý
y
`
xn Ý ns0 Ž a q b y 1, n q 1 . n!
xn
Ž a y 1, n q 1 . Ž b y 1, n q 1 . n x Ž a q b y 1, n q 1 . n!
Ž a y 1, n . Ž b y 1, n .
Ý ns0 Ž a q b y 1, n q 1 . n!
nŽ a q b y 1 q n.
yŽ a y 1 q n. Ž b y 1 q n. x n s
`
Ž a y 1, n . Ž b y 1, n .
Ý ns0 Ž a q b y 1, n q 1 . n!
n y Ž a y 1. Ž b y 1. x n .
Since Ž a y 1.Ž b y 1. F 1, the monotoneity and convexity of h now follow.
224
ANDERSON, QIU, AND VUORINEN
2.15. LEMMA. For a, b, c, d g Ž0, `. with a F b and c - d, let d 1 and d 2 be defined as in Theorem 1.7. Then
d 1 - d 2 - a.
Ž 2.16.
Proof. Since the function hŽ x . ' x 2 q Ž b y a. x y abŽ1 y Ž crd .. is strictly increasing on the real axis with hŽ d 2 . s 0 s hŽ a. y abcrd, the second inequality in Ž2.16. holds, and the first inequality in Ž2.16. holds if and only if hŽ d1 . - 0
Ž 2.17.
for all a, b, c, d g Ž0, `. with a F b and c - d. Since, by Ž1.8. and Ž2.2., c d
s
B Ž a, b . B Ž a y d1 , b q d1 .
s
G Ž a. G Ž b . G Ž a y d1 . G Ž b q d1 .
,
Ž2.17. holds if and only if G Ž a y d 1 . G Ž b q d 1 . d 12 q Ž b y a . d 1 y ab - yabG Ž a . G Ž b . . Ž 2.18. Since d 12 q Ž b y a. d 1 y ab s yŽ a y d 1 .Ž b q d 1 ., from Ž2.6. we see that Ž2.18. can be written as G Ž a q 1 y d 1 . G Ž b q 1 q d 1 . ) G Ž a q 1. G Ž b q 1. , which is true by Lemma 2.11. For our final lemma we recall a monotone version of l’Hopital’s Rule w5, ˆ x Lemma 2.2 that will be useful in proving the quotient of two functions to be monotone. 2.19. LEMMA. For y` - a - b - ` let f, g : w a, b x ª R be continuous functions, differentiable on Ž a, b ., such that g 9Ž x . / 0 for x g Ž a, b .. If f 9Ž x .rg 9Ž x . is Ž strictly . increasing Ž or Ž strictly . decreasing . on Ž a, b ., then so is Ž f Ž x . y f Ž c ..rŽ g Ž x . y g Ž c .., where c s a or b.
3. PROOFS OF THE MAIN RESULTS In this section we prove Theorems 1.7 and 1.13 and Corollary 1.19. 3.1. Proof of Theorem 1.7. Let u s a y d , ¨ s b q d s p y u, and t s 1 y Ž1 y x . d r c. We note first, by Lemma 2.11, that d 1 is uniquely determined by Ž1.8., and that d 1 , d 2 satisfy Ž2.16., by Lemma 2.15.
225
PRECISE ESTIMATES FOR DIFFERENCES
For part Ž1. we let h1 Ž x . ' h Ž Ž 1 y x .
1rc
1
.s
x
F Ž u, ¨ ; p; t . y F Ž a, b; p; x . . Ž 3.2.
Let f Ž x . s F Ž u, ¨ ; p; t . y F Ž a, b; p; x ., and g Ž x . s x. Then g Ž0. s 0, fXŽ x. g 9Ž x .
u¨ d
s f 9Ž x . s
cp y
Ž1 y x.
ab p
Ž drc .y1
f Ž0. s
F Ž u q 1, ¨ q 1; p q 1; t .
F Ž a q 1, b q 1; p q 1; x . ,
Ž 3.3.
and f0 Ž x. s
u¨ d cp q y
ž
1y
d c
/Ž
1 y x.
Ž drc .y2
u¨ Ž u q1 . Ž ¨ q1 . d 2 p Ž p q1 . c
2
ab Ž a q 1 . Ž b q 1 . p Ž p q 1.
F Ž u q 1, ¨ q 1; p q 1; t .
Ž 1 yx . 2
wŽ drc .y1 x
F Ž u q2, ¨ q 2; p q 2; t .
F Ž a q 2, b q 2; p q 2; x . .
Ž 3.4.
We wish to show that h1Ž x . is strictly increasing on Ž0, 1.. Since h1Ž x . s f Ž x .rg Ž x ., the desired monotoneity will follow from Lemma 2.19 if we prove that f 0 Ž x . ) 0 on Ž0, 1. so that f 9Ž x .rg 9Ž x . s f 9Ž x . is increasing there. By expressing the hypergeometric functions in Ž3.4. in terms of power series and comparing coefficients, we are able to prove that f 0 Ž x . is indeed positive. The limiting values of h will be determined separately, by means of Lemma 2.11 and l’Hopital’s Rule. ˆ Let h2 Ž t . s a
u¨ d cp
q
ž
1y
d c
/Ž
1 y t . F Ž u q 1, ¨ q 1; p q 1; t .
u¨ Ž u q 1 . Ž ¨ q 1 . d 2 p Ž p q 1. c
2
2 Ž 1 y t . F Ž u q 2, ¨ q 2; p q 2; t . .
Then it follows from Ž3.4. that 2 Ž1 y x. f 0 Ž x.
s h2 Ž t . y
ab Ž a q 1 . Ž b q 1 . p Ž p q 1.
2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . .
Ž 3.5.
226
ANDERSON, QIU, AND VUORINEN
Using the series expansion for F Ž a, b; c; x ., we get h2 Ž t . s
2
d
Ž1 y t .
ž / c
ž
c d
y1
`
/ÝŽ ns0
qŽ 1 y t .
`
Ý ns0
s
2
d
Ž1 y t .
ž / c
`
Ý ns0
u, n q 1 . Ž ¨ , n q 1 .
Ž p, n q 1 . n!
tn
Ž u, n q 2 . Ž ¨ , n q 2 . n t Ž p, n q 2 . n!
Ž u, n q 1 . Ž ¨ , n q 1 . Ž p, n q 2 . n!
ž
c d
y 1 Ž p q n q 1.
/
q Ž u q n q 1. Ž ¨ q n q 1. y n Ž p q n q 1. t n s
d
2
ž / c
Ž1 y t .
`
Ý ns0
c Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q Ž n q p q 1 . t n , d Ž p, n q 2 . n! Ž 3.6.
while 2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . s F Ž a, b; p q 2; x .
Ž 3.7.
by w11, Theorem 21, p. 60x. From Ž3.7., we see that Ž1 y x . 2 F Ž a q 2, b q 2; p q 2; x . is strictly increasing in x on Ž0, 1., while trx s w1 y Ž1 y x . d r c xrx is strictly decreasing in x from Ž0, 1. onto Ž1, drc., by Lemma 2.19. Thus, since t ) x, the monotoneity of Ž3.7. gives 2 2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . - Ž 1 y t . F Ž a q 2, b q 2; p q 2; t . . Ž 3.8.
Hence, it follows from Ž3.5., Ž3.6., and Ž3.8. that
Ž1 y x.
2
f0 Ž x.
1yt )
d
ž /
y
c
2
`
Ý ns0
c Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q Ž n q p q 1 . t n d Ž p, n q 2 . n!
ab Ž a q 1 . Ž b q 1 . p Ž p q 1.
Ž 1 y t . F Ž a q 2, b q 2; p q 2; t .
227
PRECISE ESTIMATES FOR DIFFERENCES
s
ž /
ns0
`
Ž a, n q 1 . Ž b, n q 1 . Ž n q ab q p q 1 . t n Ž p, n q 2 . n!
Ý ns0
s
`
2
d
c Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q Ž n q p q 1 . t n d Ž p, n q 2 . n!
Ý
c
y
`
2
d
ž /
Ž a, n q 1 . Ž b, n q 1 . Ž n q ab q p q 1 . Ž p, n q 2 . n!
Ý
c
ns0
¡ c =~y ž / ¢ d s
ž /
ns1
¡ c =~y ž / ¢ d s
d
ž / c
ns1
½
n
c
Ž n q p. d Ž a, n . Ž b, n . Ž n q ab q p .
q
¦¥ §t
ny 1
Ž a, n . Ž b, n . Ž n q ab q p . Ž p, n q 1 . Ž n y 1 . !
Ý
= QŽ n.
Ž u, n . Ž ¨ , n . u¨ q
2
`
2
¦¥ §t
Ž a, n . Ž b, n . Ž n q ab q p . Ž p, n q 1 . Ž n y 1 . !
Ý
c
c
Ž n q p q 1. d Ž a, n q 1 . Ž b, n q 1 . Ž n q ab q p q 1 .
q
`
2
d
Ž u, n q 1 . Ž ¨ , n q 1 . u¨ q
2
G Ž a. G Ž b . G Ž u. G Ž ¨ .
y
2
c
ž /5 d
t ny 1 ,
Ž 3.9.
where QŽ n. is as in Lemma 2.9. Hence, it follows from Lemma 2.9 and Ž3.9. that
Ž 1 y x . 2y ) s
Ž drc .
f0 Ž x.
d G Ž a. G Ž b . c G Ž u. G Ž ¨ . 1
d
B Ž u, ¨ .
c
=
`
Ý ns1
y
c d
`
Ý ns1
Ž a, n . Ž b, n . Ž n q ab q p . ny 1 t Ž p, n q 1 . Ž n y 1 . !
B Ž a, b . y B Ž u, ¨ .
Ž a, n . Ž b, n . Ž n q ab q p . ny 1 t . Ž p, n q 1 . Ž n y 1 . !
Ž 3.10.
By Lemma 2.11, as a function of d , Ž drc. B Ž a, b . y B Ž u, ¨ . is strictly decreasing on Ž0, a.. Since 0 - d F d 1 , we have d d B Ž a, b . y B Ž u, ¨ . G B Ž a, b . y B Ž a y d 1 , b q d 1 . s 0. c c
228
ANDERSON, QIU, AND VUORINEN
Hence, it follows from Ž3.10. that f 0 Ž x . ) 0 for all x g Ž0, 1.. This shows that f 9Ž x . is strictly increasing on Ž0, 1., and so is h1Ž x . by Ž3.2., Ž3.3., and Lemma 2.19. The asserted monotoneity of h now follows. By l’Hopital’s Rule, Ž3.2., and Ž3.3. we get ˆ u¨ d
h Ž 1y . s h1 Ž 0q . s f 9 Ž 0 . s s
d cp
ab 1 y
½ ž
c d
cp
y
ab p
y d 2 q Ž b y a. d
/
5
s A1 Ž d . . Ž 3.11.
By Lemma 2.15, for 0 - d F d 1 , we have A1Ž d . G A1Ž d 1 . ) A1Ž d 2 . s 0. On the other hand, since F Ž a, b; a q b; x . s
`
1
Ž a, n . Ž b, n .
Ý B Ž a, b . ns0
Ž n! .
2
2C Ž n q 1 . y C Ž a q n .
yC Ž b q n . y log Ž 1 y x . Ž 1 y x .
n
Ž 3.12.
by w1, 15.3.10x, and, since C Ž1. s yg w1, 6.3.2x, we have h Ž 0q . s h1 Ž 1y . s f Ž 1y . s limy xª1
1
½
B Ž u, ¨ . y
s limy xª1
½
R Ž u, ¨ . y log Ž 1 y t .
1
R Ž a, b . y log Ž 1 y x .
B Ž a, b .
R Ž u, ¨ . B Ž u, ¨ . =
y d c
R Ž a, b . B Ž a, b .
q
5
1 B Ž u, ¨ . B Ž a, b .
B Ž a, b . y B Ž u, ¨ . log
1 1yx
s A2 Ž d . .
5 Ž 3.13.
Here, in the last equality, we have used Lemma 2.11. Formula Ž1.11. and the first inequality in Ž1.12. are clear, while the other inequalities in Ž1.12. and the equality case follow from Lemma 2.13Ž2.. For part Ž2., we observe that, for d 1 - d - d 2 , Ž3.11. and the first four equalities in Ž3.13. still hold. Since A1Ž d . is strictly decreasing in d , h Ž 1y . s A1 Ž d . ) A1 Ž d 2 . s 0,
d1 - d - d2 .
229
PRECISE ESTIMATES FOR DIFFERENCES
By Lemma 2.11, d c
B Ž a, b . y B Ž u, ¨ . -
d c
B Ž a, b . y B Ž a y d 1 , b q d 1 . s 0
for all d 1 - d - d 2 , so that from Ž3.13. we get h Ž 0q . s y`,
d1 - d - d2 .
Hence, part Ž2. follows. 3.14. COROLLARY. Under the hypotheses of Theorem 1.7Ž1., the function f Ž x . ' F Ž a y d , b q d ; a q b; 1 y Ž 1 y x .
drc
. y F Ž a, b; a q b; x .
is strictly increasing and con¨ ex from Ž0, 1. onto Ž0, A 2 ., where A 2 s A 2 Ž d . is as in Theorem 1.7. Proof. In the proof of Theorem 1.7Ž1. we showed that f 9Ž x . is positive and strictly increasing on Ž0, 1. and that f Ž0. s f Ž1y . y A 2 s 0. 3.15. Proof of Theorem 1.13. As in the proof of Theorem 1.7, let u s a y d , ¨ s b q d s p y u, t s 1 y Ž1 y x . d r c, and let h1 Ž x . s
1 c
hŽ Ž1 y x .
1rc
.s
F Ž u, ¨ ; p; t . y F Ž a, b; p; x . log Ž 1r Ž 1 y x . .
.
In order to prove the first statement, we need only show that h1 is strictly decreasing from Ž0, 1. onto Ž A 3rc, A 4rc . if d 2 F d - a. For this, in the notation of Lemma 2.19 let f Ž x . s F Ž u, ¨ ; p; t . y F Ž a, b; p; x . ,
g Ž x . s log
1 1yx
.
Then f Ž0. s g Ž0. s 0 and fXŽ x. X
g Ž x.
s h2 Ž x . ' Ž 1 y x .
u¨ d cp
Ž1 y x.
Ž drc .y1
y s
u¨ d cp y
ab p
F Ž u q 1, ¨ q 1; p q 1; t .
F Ž a q 1, b q 1; p q 1; x .
Ž 1 y t . F Ž u q 1, ¨ q 1; p q 1; t .
ab p
Ž 1 y x . F Ž a q 1, b q 1; p q 1; x . .
Ž 3.16.
230
ANDERSON, QIU, AND VUORINEN
Since h1Ž x . s f Ž x .rg Ž x ., the desired monotoneity will follow from Lemma 2.19 if we prove that h 2 Ž x . is decreasing. As in the proof of Theorem 1.7 we shall do this by expressing the hypergeometric functions in Ž3.16. in terms of power series and comparing coefficients. By Ž3.16. and l’Hopital’s Rule, we get ˆ h1 Ž 0q . s h 2 Ž 0 . s
u¨ d cp
ab
y
p
s
A4 c
,
which is nonpositive by the definition of d 2 and the monotoneity of A 4 Ž d .. On the other hand, Ž3.12. yields h1 Ž 1y . s lim
xª1
1 log Ž 1r Ž 1 y x . .
½
1 B Ž u, ¨ . 1
y s
d
y
cB Ž u, ¨ .
1 B Ž a, b .
s
B Ž a, b . A3 c
d
R Ž u, ¨ . y
c
log Ž 1 y x .
R Ž a, b . y log Ž 1 y x .
5
.
Next, differentiation Žsee Ž3.16.. gives
Ž 1 y x . hX2 Ž x . s h 3 Ž x . ' y y
pc 2
Ž 1 y t . F Ž u q 1, ¨ q 1; p q 1; t .
Ž u q 1. Ž ¨ q 1. pq1 q
y
u¨ d 2
Ž a q 1. Ž b q 1. pq1
ab p
2 Ž 1 y t . F Ž u q 2, ¨ q 2; p q 2; t .
Ž 1 y x . F Ž a q 1, b q 1; p q 1; x .
2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . . Ž 3.17.
Since p q 1 s Ž u q 1. q Ž ¨ q 1. y 1, and since u¨ s y d 2 q Ž b y a . d y ab 1 y
ž
c d
F y d 22 q Ž b y a . d 2 y ab 1 y
ž
s
abc d
F 1,
q
/
c d
/
abc
q
d abc d
Ž 3.18.
PRECISE ESTIMATES FOR DIFFERENCES
231
it follows from Lemma 2.14 that the expression in the first set of brackets in Ž3.17. is strictly increasing in t on Ž0, 1.. Hence, since x - t, u¨ d 2
h3 Ž x . - y
Ž 1 y x . F Ž u q 1, ¨ q 1; p q 1; x .
pc 2
Ž u q 1. Ž ¨ q 1.
y q
ab
pq1
2 Ž 1 y x . F Ž u q 2, ¨ q 2; p q 2; x .
Ž 1 y x . F Ž a q 1, b q 1; p q 1; x .
p
Ž a q 1. Ž b q 1.
y
pq1
2 Ž 1 y x . F Ž a q 2, b q 2; p q 2; x . ,
so that Ž3.17. and Ž2.6. yield
hX2 Ž x . - y
u¨ d 2 pc 2
F Ž u q 1, ¨ q 1; p q 1; x . y
ab
q
p
d2 c
2
pq1
`
Ý ns0
Ž a q 1. Ž b q 1. pq1
`
Ý ns0
`
Ý ns0
Ž 1 y x . F Ž a q 2, b q 2; p q 2; x .
Ž u, n q 1 . Ž ¨ , n q 1 . n x Ž p, n q 1 . n!
yŽ 1 y x . q
Ž 1 y x . F Ž u q 2, ¨ q 2; p q 2; x .
F Ž a q 1, b q 1; p q 1; x . y
sy
Ž u q 1. Ž ¨ q 1.
Ž u, n q 2 . Ž ¨ , n q 2 . n x Ž p, n q 2 . n!
Ž a, n q 1 . Ž b, n q 1 . n x Ž p, n q 1 . n!
yŽ 1 y x .
`
Ý ns0
Ž a, n q 2 . Ž b, n q 2 . n x Ž p, n q 2 . n!
232
ANDERSON, QIU, AND VUORINEN
sy
`
2
d
ž /
Ý
c
ns0
Ž u, n q 1 . Ž ¨ , n q 1 . Ž p, n q 2 . n!
= Ž n q 1. Ž p q n q 1. y Ž u q n q 1. Ž ¨ q n q 1. x n `
q
Ž a, n q 1 . Ž b, n q 1 . Ž n q 1. Ž p q n q 1. Ž p, n q 2 . n!
Ý ns0
y Ž a q n q 1. Ž b q n q 1. x n s
d
ž /
Ý
c
y
`
2
u¨ Ž u, n q 1 . Ž ¨ , n q 1 .
Ž p, n q 2 . n!
ns0
`
ab Ž a, n q 1 . Ž b, n q 1 .
Ý
Ž p, n q 2 . n!
ns0
s s
`
Ž a, n . Ž b, n .
u¨
Ý ns1 Ž p, n q 1 . Ž n y 1 . ! `
xn
xn 2
d
ž / ŽŽ c
u, n . Ž ¨ , n . a, n . Ž b, n .
y ab x ny 1
Ž a, n . Ž b, n .
Ý ns1 Ž p, n q 1 . Ž n y 1 . !
= u¨
2
d
ž /
G Ž n q u . G Ž n q ¨ . G Ž a. G Ž b . G Ž n q a. G Ž n q b . G Ž u . G Ž ¨ .
c
y ab x ny 1 .
Thus it follows from Lemma 2.10 and Ž2.6. that hX2
Ž x. -
d
1
2
ž /
ab
c
2
Ž u¨ . y
abc
`
2
ž /
Ž a, n . Ž b, n .
x ny1 , Ý ns1 Ž p, n q 1 . Ž n y 1 . !
d
which is nonpositive by Ž3.18.. Hence, h 2 is strictly decreasing on Ž0, 1.. The monotoneity of h1 now follows from Lemma 2.19 and Ž3.16.. The first inequality in Ž1.17. follows from Lemma 2.13Ž2., while Ž1.16., the other parts of Ž1.17., and the equality case are clear. 3.19. Proof of Corollary 1.19. For part Ž1., we observe that if a s b s 1r2 in Theorem 1.7, then d 1 satisfies cB
ž
1 2
y d1 ,
1 2
q d 1 s dB
/
1 1 , 2 2
ž /
or, equivalently, cG
ž
1 2
y d1 G
/ž
1 2
q d 1 s dG
/
1
ž / 2
2
s p d,
Ž 3.20.
233
PRECISE ESTIMATES FOR DIFFERENCES
so that, by Ž2.7.,
d1 s
1
p
arccos
c
ž / d
,
since Ž1r2. q d 1 s 1 y wŽ1r2. y d 1 x. Clearly, d 2 s Next, it follows from Ž2.5., Ž3.20., and Ž1.10. that A2 Ž d 1 . s
c
1
p d
R
ž
1 2
y d1 ,
1 2
1 2
'1 y Ž crd . .
q d 1 y log 16 ,
/
while Ž1.20. holds by Ž1.9.. Part Ž1. now follows from Theorem 1.7Ž1.. By Theorem 1.7Ž2., part Ž3. is clear. For part Ž2., by Theorem 1.13 we need only verify that ab F drc for a s b s 1r2. But this is clear. 3.21. Remark. Ž1. In Theorems 1.7 and 1.13, if a q b s p s 1, then, by Ž2.7.,
¡B Ž a y d , b q d . s G Ž a y d . G Ž 1 y Ž a y d . . s
~
¢
B Ž a, b . s G Ž a . G Ž 1 y a . s
p sin ap
p sin Ž a y d . p
,
,
and hence d 1 has the explicit expression
d1 s
1
p
½
ap y arcsin
ž
c d
sin ap
/5
say
1 2
q
1
p
arccos
ž
c d
sin ap .
/
In particular, if a s 1r2, then d 1 s Ž1rp .arccosŽ crd .. Ž2. Our computational work supports the validity of the following conjecture: The condition ‘‘abc F d’’ in Theorem 1.13 is not necessary.
ACKNOWLEDGMENTS This research was completed during the second author’s visit to Michigan State University, with grants from its Department of Mathematics and from the Academy of Finland. The authors are grateful to the referee for helpful suggestions for the improvement of the presentation.
234
ANDERSON, QIU, AND VUORINEN
REFERENCES 1. M. Abramowitz and I. A. Stegun ŽEds.., ‘‘Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,’’ Dover, New York, 1965. 2. L. V. Ahlfors, ‘‘Complex Analysis,’’ 2nd ed., McGraw]Hill, New York, 1966. 3. G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 Ž1995., 1713]1723. 4. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Hypergeometric functions and elliptic integrals, in ‘‘Current Topics in Analytic Function Theory’’ ŽH. M. Srivastava and S. Owa, Eds.., pp. 48]85, World Scientific, SingaporerLondon, 1992. 5. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for quasiconformal mappings in space, Pacific J. Math. 160 Ž1993., 1]18. 6. R. Askey, Ramanujan and hypergeometric and basic hypergeometric series, in ‘‘Ramanujan Internat. Symposium on Analysis, December 26]28, 1987’’ ŽN. K. Thakare, Ed.., Pune, India. 7. B. C. Berndt, ‘‘Ramanujan’s Notebooks,’’ Part II, Springer-Verlag, BerlinrHeidelbergrNew York, 1989. 8. B. C. Berndt, S. Bhargava, and F. G. Garvan, Ramanujan’s theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347 Ž1995., 4163]4244. 9. J. M. Borwein and P. B. Borwein, Inequalities for compound mean iterations with logarithmic asymptotes, J. Math. Anal. Appl. 177 Ž1993., 572]582. 10. S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 Ž1997., 278]301. 11. E. D. Rainville, ‘‘Special Functions,’’ Macmillan, New York, 1960. 12. E. T. Whittaker and G. N. Watson, ‘‘A Course of Modern Analysis,’’ Cambridge Univ. Press, Cambridge, 1958.