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Some Mean Values Related to the Arithmetic–Geometric Mean
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
218, 358]368 Ž1998.
AY975766
Some Mean Values Related to the Arithmetic]Geometric Mean Gh. Toader Department of Mathematics, Technical Uni¨ ersity, RO-3400 Cluj-Napoca, Romania Submitted by A. M. Fink Received March 17, 1997
1. INTRODUCTION Let a, b be positive real numbers. The arithmetic]geometric mean of a and b is defined as the common limit UŽ a, b . of the sequences Ž a n ., Ž bn . given recurrently by a0 s a,
b 0 s b,
a nq1 s Ž a n q bn . r2,
bnq1 s a n bn .
'
The mean U was considered firstly by Lagrange but its connection with elliptic integrals is due to Gauss. For more historical remarks on U, see w2, 6x. In what follows we also use the power means defined by Pq Ž a, b . s
ž
aq q b q 2
1rq
/
for q / 0, while, for q s 0, P0 Ž a, b . s G Ž a, b . s 'ab is the geometric mean. As A s P1 , the harmonic mean Generally, a mean value which satisfies the following
special cases we have: the arithmetic mean H s Py1 , and the square-root mean Q s P2 . is defined as a function M: Rq= Rqª Rq postulate
min Ž a, b . F M Ž a, b . F max Ž a, b . , 358 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
; a, b ) 0.
Ž 1.
359
SOME MEAN VALUES
This definition was given in essence by Cauchy Žsee w3x. and it remains the most used though some other definitions were proposed. Of course, a mean has the reflexivity property M Ž a, a . s a,
; a ) 0.
Ž 2.
The mean is called symmetric if M Ž a, b . s M Ž b, a . ,
; a, b ) 0.
Ž 3.
Properties Ž2. and Ž3. have been used in w1x to define a mean Žsee also w4, 5x.. Let us denote rn Ž u . s Ž a n cos 2u q b n sin 2u .
1rn
n / 0,
,
and r 0 Ž u . s lim rn Ž u . s acos u b sin u . 2
2
nª0
For a strictly monotonic function p: Rqª R, we set M p , n Ž a, b . s py1
ž
1
2p
H 2p 0
p Ž r n Ž u . . du .
/
Ž 4.
It is easy to prove that M p, n is a mean value. In fact, it is a special case of the integral quasi-arithmetic mean Žsee w2, p. 379x.. As M p , n Ž a, b . s py1 s py1
ž ž
1
p
H p 0 2
p
p Ž r n Ž u . . du
pr2
H0
/
p Ž r n Ž u . . du ,
/
Ž 49 .
we note that M p, n is symmetric. So that in what follows we can suppose a G b. For the case n s 2, in w4x are characterized the functions p for which M p, n is one of the following means: U, A, G, or Q. A similar study was done in w5x in the case n s y1 or n s 1. In this paper, considering the general case, we get unique proofs for arbitrary n Žthus including the above-mentioned three cases.. Also we show that it is more natural to consider the results as simply giving another concrete expression of M p, n in some special cases than as a characterization of some means. We continue this attempt by indicating a possibility of getting other such expressions using recurrence formulas.
360
GH. TOADER
2. A FUNCTIONAL EQUATION Let us consider the function f Ž a, b; p, n . s
1 2p
2p
H0
p Ž r n Ž u . . du .
Ž 5.
This is well known in the case n s 2 and pŽ x . s xy1 being related to the complete elliptic integral of the first kind. It is also known Žsee w4x. that in this case M p, n s U, the arithmetic]geometric mean. The proof follows from the fact that the function G is a solution of the functional equation: f Ž A Ž a, b . , G Ž a, b . . s f Ž a, b . . This equation was studied in w4x. Here we consider a more general functional equation f Ž Pq Ž a, b . , Ps Ž a, b . . s f Ž a, b . .
Ž 6.
Denoting the power function en Ž x . s x n
for n / 0
and e0 Ž x . s log x, we shall prove the following theorem. THEOREM 1. If the function f is a solution of Ž6. which can be represented by Ž5., where p has a continuous second-order deri¨ ati¨ e in Rq, then p s a e qq syn q b ,
Ž 7.
with a , b arbitrary real constants. Remark. For n s 2, q s 1, and s s 0, we get the necessity part of Theorem 1 of w4x. In this case, as we have already mentioned, the condition is also sufficient. In fact, it is the only case with this property that we know. To prove the above theorem, we need the following lemma. LEMMA. Let p: Rqª R be twice continuously differentiable in Rq. Then the function f defined by Ž5. has the following partial deri¨ ati¨ es:
Ž i.
f a Ž c, c; p, n . s f b Ž c, c; p, n . s p9 Ž c . r2,
Ž ii .
f aa Ž c, c; p, n . s 3cp0 Ž c . q Ž n y 1 . p9 Ž c . r8c,
Ž iii .
f ab Ž c, c; p, n . s cp0 Ž c . y Ž n y 1 . p9 Ž c . r8c,
where c is an arbitrary positi¨ e real number.
361
SOME MEAN VALUES
Proof. We follow the lines of the proof of a similar lemma from w4x. We apply differentiation under the integral sign. From Ž5. we obtain 1
f a Ž a, b; p, n . s
2p
2p
H0
p9 Ž rn Ž u . .
ž
ny 1
a
cos 2u du .
/
rn Ž u .
Ž 8.
Setting a s b s c, we get Ži.. By Ž8. we also obtain f aa Ž a, b; p, n . s
Ž n y 1 . a ny 2 2p q
a ny 1
2p
H0
2p =
ž
2p
H0
p9 Ž rn Ž u . . rn1y n Ž u . cos 2u du
p0 Ž rn Ž u . . rn1y n Ž u . q p9 Ž rn Ž u . . Ž 1 y n . ryn n Žu . ny 1
a rn Ž u .
cos 4 Ž u . du .
/
Setting here a s b s c yields Žii.. In a similar manner we can prove Žiii.. Proof of Theorem 1. Applying r a to both sides of Ž6. yields f a Ž a, b; p, n . s f a Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
a qy1 2
q f b Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
aq q b q
ž
2
a sy1 2
1rqy1
/
as q b s
ž
2
1rsy1
/
Applying to this r b, we get f ab Ž a, b; p, n . s f aa Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
b qy1 2
q f ab Ž Pq Ž a, b . , Ps Ž a, b . ; p, n . = =
a qy1 2
ž
aq q b q 2
2
b sy1 2
2
ž
1rqy1
/
as q b s 2
1rsy1
/
1rqy1
q f a Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
/
a qy1 Ž 1 y q . b qy1 2
ž
aq q b q
ž
aq q b q 2
1rqy2
/
q f ab Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
b qy1 2
ž
aq q b q 2
1rqy1
/
.
362
GH. TOADER
q f b b Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
= =
a sy1 2
as q b s
ž
2
2
2
ž
as q b s 2
1rsy1
/
1rsy1
q f b Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
/
a sy1 Ž 1 y s . b sy1 2
b sy1
ž
as q b s 2
1rsy2
/
.
Setting a s b s c yields f aa Ž c, c; p, n . y 2 f ab Ž c, c; p, n . q f b b Ž c, c; p, n . q
1yq c
f a Ž c, c; p, n . q
1ys c
f b Ž c, c; p, n . s 0.
Using Ži., Žii., and Žiii., we get cp0 Ž c . q Ž n y q y s q 1 . p9 Ž c . s 0, with solution Ž7.. We remark that the results are also valid if one of the parameters q and s vanishes.
3. POWER MEANS As was shown in w4, 5x, the means M p, n can represent some known means for special choices of p and n. In this section we want to determine the functions p for which M p, n is a power mean Pq , where n and q are arbitrary real numbers. THEOREM 2. If for some twice continuously differentiable function p the mean M p, n reduces at the power mean Pq , then p s a e2 qyn q b , where a and b are arbitrary real numbers. Proof. Using Ž4. and Ž5., we have M p , n Ž a, b . s Pq Ž a, b . if and only if f Ž a, b; p, n . s p Ž Pq Ž a, b . . .
Ž 79 .
363
SOME MEAN VALUES
Applying 2r a b to both sides, we get
f ab Ž a, b; p, n . s p0 Ž Pq Ž a, b . .
ž
aq q b q
q p9 Ž Pq Ž a, b . .
2
2 Ž1rqy1 .
/
b qy1 a qy1 2
a qy1 Ž 1 y q . b qy1 2
2
2
ž
aq q b q 2
1rqy2
/
.
Setting a s b s c and using the lemma yields cp0 Ž c . y Ž n y 1 . p9 Ž c . r2 c s p0 Ž c . q p9 Ž c . Ž 1 y q . rc. Thus cp0 Ž c . s Ž 2 q y n y 1 . p9 Ž c . , with the solution Ž79.. Remark. For n s 2 and q s 0, 1, and 2, the result was proved in w4x. For n s 1 and q s 0, 1, and 1r2 or n s y1 and q s y1, y1r2, and 0, it was proved in w5x. In all these cases the condition was proved to be also sufficient, that is, M p, n s Pq . But this is not generally valid. For example, in the case n s 2 and p s ey1 , we have M p, n s U and not P1r2 . Also M p, 2 / Py1 s H for p s e4 as was proved in w4x. Other cases in which M p, n / Pq for p s e2 qyn were given in w5x as we shall show in what follows. 4. A RECURRENCE FORMULA In what follows we use f Ž a, b; p, n. and M p, n only for p s e q , so that we denote them by f Ž a, b; q, n. and Mq, n , respectively. Also, as Mq , n Ž a, b . s f Ž a, b; q, n .
1rq
for q / 0
and M0, n Ž a, b . s exp f Ž a, b; 0, n . , we usually content ourselves with determining the function f for some values of q and n.
364
GH. TOADER
In w4, 5x the following values are given: f Ž a, b; 0, 2 . s log 1
f Ž a, b; y2, 2 . s
ab
aqb 2 ,
a2 q b 2
f Ž a, b; 2, 2 . s
2 1
f Ž a, b; y2, 1 . s
ž
2'ab 1
f Ž a, b; y1, 1 . s
ž
, 1 a
q
1
/
b
,
Ž 9.
,
'ab
f Ž a, b; 0, 1 . s log
,
Ž 10 . 2
a1r2 q b1r2
/
2
.
As we have already mentioned, 2
f Ž a, b; y1, 2 . s
pa
K
ž(
1y
2
b
ž // a
,
Ž 11 .
where K is the complete elliptic integral of the first kind. Similarly, f Ž a, b; 1, 2 . s
2a
p
E
ž(
1y
b
2
ž // a
,
Ž 12 .
where E is the complete elliptic integral of the second kind. To obtain new formulas, let us begin by remarking that f Ž a, b; q, n . s f Ž a n , b n ; qrn, 1 . .
Ž 13 .
Thus, for the moment, it is enough to consider the case n s 1. As in Ž49. we have f Ž a, b; q, 1 . s s
2
p 2
p
pr2
H0
ž
aqb 2
q
Ž a cos 2u q b sin 2u . du q
/
pr2
H0
ž
1q
ayb aqb
q
cos 2 u
/
du .
365
SOME MEAN VALUES
If we denote Jq s
pr2
H0
q
Ž 1 q k cos 2 u . du
and
2
aqb
ks
ayb aqb
,
we get f Ž a, b; q, 1 . s
p
ž
2
q
/
Jq .
Ž 14 .
We look for a recurrence formula for Jq . We have Jq s
pr2
H0
Ž 1 q k cos 2 u . Ž 1 q k cos 2 u .
s Jqy 1 q s Jqy1 q q
pr2
H0
k
pr2
H 2 0 k 2
qy 1
Ž sin 2 u . 9 Ž 1 q k cos 2 u .
sin 2 u Ž 1 q k cos 2 u .
du qy 1
qy 1 p r2 <0
sin 2 2 u Ž q y 1 . k 2 Ž 1 q k cos 2 u .
s Jqy 1 q k 2 Ž q y 1 .
pr2
H0
du
qy 2
du
Ž 1 y cos 2 2 u . Ž 1 q k cos 2 u .
qy 2
du
s Jqy 1 q k 2 Ž q y 1 . Jqy2 y Ž q y 1 . =
pr2
H0
2 Ž k cos 2 u q 1 . y 2 Ž k cos 2 u q 1 . q 1
= Ž 1 y k cos 2 u .
qy 2
du
s Jqy 1 q k 2 Ž q y 1 . Jqy2 y Ž q y 1 . Ž Jq y 2 Jqy1 q Jqy2 . . Thus qJq s Ž 2 q y 1 . Jqy1 q Ž q y 1 . Ž k 2 y 1 . Jqy2 . Of course, q, q y 1, and q y 2 are different from 0. As k2 y 1 s
y4 ab
Ž a q b.
2
,
Ž 15 .
366
GH. TOADER
the relations Ž14. and Ž15. give qf Ž a, b; q, 1 . y Ž q y 1r2 . Ž a q b . f Ž a, b; q y 1, 1 . q ab Ž q y 1 . f Ž a, b; q y 2, 1 . s 0.
Ž 16 .
Replacing a and b by a n and b n, respectively, and rewriting Ž13. as f Ž a n , b n ; q, 1 . s f Ž a, b; nq, n . ,
Ž 139.
we have qf Ž a, b; nq, n . y Ž q y 1r2 . Ž a n q b n . f Ž a, b; n Ž q y 1 . , n . q a n b n Ž q y 1 . f Ž a, b; n Ž q y 2 . , n . s 0. Using these relations, starting from two adequate values of q, we can obtain successively more values of f Ž a, b; q, n.. For example, using Ž9., Ž10., and Ž16., we get f Ž a, b; y3, 1 . s
3a2 q 2 ab q 3b 2 8 Ž ab .
5r2
.
Then f Ž a, b; y4, 1 . s
5a3 q 3a2 b q 3ab 2 q 5b 3 16 Ž ab .
7r2
and so on. Also it is easy to see that f Ž a, b; 1, 1 . s
aqb 2
and f Ž a, b; 2, 1 . s
3a2 q 2 ab q 3b 2 8
.
From Ž16. we have f Ž a, b; 3, 1 . s
5a3 q 3a2 b q 3ab 2 q 5b 3 16
.
367
SOME MEAN VALUES
Then f Ž a, b; 4, 1 . s
35a4 q 20 a3 b q 18 a2 b 2 q 20 ab 3 q 35b 4 128
and so on. Using Ž13., we can rewrite Ž11. and Ž12. as f a, b; y
ž
1 2
, 1 s f Ž 'a , 'b ; y1, 2 . s
/
2
K
p'a
b
ž( / 1y
a
and
ž
f a, b;
1 2
2'a
, 1 s f Ž 'a , 'b ; 1, 2 . s
/
p
E
b
ž( / 1y
a
.
Taking in Ž16. q s 3r2, we get
ž
f a, b;
3 2
,1 s
/
2'a 3p
2Ž a q b . E
b
b
ž( / ž( / 1y
y bK
a
1y
a
.
Then, for q s 1r2, f a, b; y
ž
3 2
2
,1 s
/
p b'a
E
b
ž( / 1y
a
and so on, we can express any f Ž a, b; Ž2 q q 1.r2, 1. by means of K and E. In fact, for n s 1, we are able to determine any function f Ž a, b; q, 1. for q s k or q s Ž2 k q 1.r2, with k integer, but only for such values of q. All these functions can be transcribed by Ž139. to give their expressions for an arbitrary n / 0: f Ž a, b; nq, n . s f Ž a n , b n ; q, 1 . . We reobtain all the means given in w4, 5x, but we can also determine many other means Mq, n . For n s 0, it is easy to prove that M0, 0 s G, but for q / 0 we do not know how to determine any mean Mq , 0 Ž a, b . s
ž
1
2p q cos 2 u
H 2p 0
a
1rq
b q sin
2
u
du
/
.
368
GH. TOADER
REFERENCES 1. J. Aczel, ´ The notion of mean value, Norske Vid. Selsk. Forh. ŽTrondheim. 19 Ž1946., 83]86. 2. P. S. Bullen, D. S. Mitrinovic, ´ and P. M. Vasic, ´ ‘‘Means and Their Inequalities,’’ Reidel, Dordrecht, 1988. 3. C. Gini, ‘‘Means,’’ Unione Tipografico-Editrice Torinese, Milan, 1958. wIn Italian x 4. H. Haruki, New characterizations of the arithmetic]geometric mean of Gauss and other well-known mean values, Publ. Math. Debrecen 38 Ž1991., 323]332. 5. H. Haruki and T. M. Rassias, New characterizations of some mean-values, J. Math. Anal. Appl. 202 Ž1996., 333]348. 6. M. K. Vamanamurthy and M. Vuorinen, Inequalities for means, J. Math. Anal. Appl. 183 Ž1994., 155]166.
218, 358]368 Ž1998.
AY975766
Some Mean Values Related to the Arithmetic]Geometric Mean Gh. Toader Department of Mathematics, Technical Uni¨ ersity, RO-3400 Cluj-Napoca, Romania Submitted by A. M. Fink Received March 17, 1997
1. INTRODUCTION Let a, b be positive real numbers. The arithmetic]geometric mean of a and b is defined as the common limit UŽ a, b . of the sequences Ž a n ., Ž bn . given recurrently by a0 s a,
b 0 s b,
a nq1 s Ž a n q bn . r2,
bnq1 s a n bn .
'
The mean U was considered firstly by Lagrange but its connection with elliptic integrals is due to Gauss. For more historical remarks on U, see w2, 6x. In what follows we also use the power means defined by Pq Ž a, b . s
ž
aq q b q 2
1rq
/
for q / 0, while, for q s 0, P0 Ž a, b . s G Ž a, b . s 'ab is the geometric mean. As A s P1 , the harmonic mean Generally, a mean value which satisfies the following
special cases we have: the arithmetic mean H s Py1 , and the square-root mean Q s P2 . is defined as a function M: Rq= Rqª Rq postulate
min Ž a, b . F M Ž a, b . F max Ž a, b . , 358 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
; a, b ) 0.
Ž 1.
359
SOME MEAN VALUES
This definition was given in essence by Cauchy Žsee w3x. and it remains the most used though some other definitions were proposed. Of course, a mean has the reflexivity property M Ž a, a . s a,
; a ) 0.
Ž 2.
The mean is called symmetric if M Ž a, b . s M Ž b, a . ,
; a, b ) 0.
Ž 3.
Properties Ž2. and Ž3. have been used in w1x to define a mean Žsee also w4, 5x.. Let us denote rn Ž u . s Ž a n cos 2u q b n sin 2u .
1rn
n / 0,
,
and r 0 Ž u . s lim rn Ž u . s acos u b sin u . 2
2
nª0
For a strictly monotonic function p: Rqª R, we set M p , n Ž a, b . s py1
ž
1
2p
H 2p 0
p Ž r n Ž u . . du .
/
Ž 4.
It is easy to prove that M p, n is a mean value. In fact, it is a special case of the integral quasi-arithmetic mean Žsee w2, p. 379x.. As M p , n Ž a, b . s py1 s py1
ž ž
1
p
H p 0 2
p
p Ž r n Ž u . . du
pr2
H0
/
p Ž r n Ž u . . du ,
/
Ž 49 .
we note that M p, n is symmetric. So that in what follows we can suppose a G b. For the case n s 2, in w4x are characterized the functions p for which M p, n is one of the following means: U, A, G, or Q. A similar study was done in w5x in the case n s y1 or n s 1. In this paper, considering the general case, we get unique proofs for arbitrary n Žthus including the above-mentioned three cases.. Also we show that it is more natural to consider the results as simply giving another concrete expression of M p, n in some special cases than as a characterization of some means. We continue this attempt by indicating a possibility of getting other such expressions using recurrence formulas.
360
GH. TOADER
2. A FUNCTIONAL EQUATION Let us consider the function f Ž a, b; p, n . s
1 2p
2p
H0
p Ž r n Ž u . . du .
Ž 5.
This is well known in the case n s 2 and pŽ x . s xy1 being related to the complete elliptic integral of the first kind. It is also known Žsee w4x. that in this case M p, n s U, the arithmetic]geometric mean. The proof follows from the fact that the function G is a solution of the functional equation: f Ž A Ž a, b . , G Ž a, b . . s f Ž a, b . . This equation was studied in w4x. Here we consider a more general functional equation f Ž Pq Ž a, b . , Ps Ž a, b . . s f Ž a, b . .
Ž 6.
Denoting the power function en Ž x . s x n
for n / 0
and e0 Ž x . s log x, we shall prove the following theorem. THEOREM 1. If the function f is a solution of Ž6. which can be represented by Ž5., where p has a continuous second-order deri¨ ati¨ e in Rq, then p s a e qq syn q b ,
Ž 7.
with a , b arbitrary real constants. Remark. For n s 2, q s 1, and s s 0, we get the necessity part of Theorem 1 of w4x. In this case, as we have already mentioned, the condition is also sufficient. In fact, it is the only case with this property that we know. To prove the above theorem, we need the following lemma. LEMMA. Let p: Rqª R be twice continuously differentiable in Rq. Then the function f defined by Ž5. has the following partial deri¨ ati¨ es:
Ž i.
f a Ž c, c; p, n . s f b Ž c, c; p, n . s p9 Ž c . r2,
Ž ii .
f aa Ž c, c; p, n . s 3cp0 Ž c . q Ž n y 1 . p9 Ž c . r8c,
Ž iii .
f ab Ž c, c; p, n . s cp0 Ž c . y Ž n y 1 . p9 Ž c . r8c,
where c is an arbitrary positi¨ e real number.
361
SOME MEAN VALUES
Proof. We follow the lines of the proof of a similar lemma from w4x. We apply differentiation under the integral sign. From Ž5. we obtain 1
f a Ž a, b; p, n . s
2p
2p
H0
p9 Ž rn Ž u . .
ž
ny 1
a
cos 2u du .
/
rn Ž u .
Ž 8.
Setting a s b s c, we get Ži.. By Ž8. we also obtain f aa Ž a, b; p, n . s
Ž n y 1 . a ny 2 2p q
a ny 1
2p
H0
2p =
ž
2p
H0
p9 Ž rn Ž u . . rn1y n Ž u . cos 2u du
p0 Ž rn Ž u . . rn1y n Ž u . q p9 Ž rn Ž u . . Ž 1 y n . ryn n Žu . ny 1
a rn Ž u .
cos 4 Ž u . du .
/
Setting here a s b s c yields Žii.. In a similar manner we can prove Žiii.. Proof of Theorem 1. Applying r a to both sides of Ž6. yields f a Ž a, b; p, n . s f a Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
a qy1 2
q f b Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
aq q b q
ž
2
a sy1 2
1rqy1
/
as q b s
ž
2
1rsy1
/
Applying to this r b, we get f ab Ž a, b; p, n . s f aa Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
b qy1 2
q f ab Ž Pq Ž a, b . , Ps Ž a, b . ; p, n . = =
a qy1 2
ž
aq q b q 2
2
b sy1 2
2
ž
1rqy1
/
as q b s 2
1rsy1
/
1rqy1
q f a Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
/
a qy1 Ž 1 y q . b qy1 2
ž
aq q b q
ž
aq q b q 2
1rqy2
/
q f ab Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
b qy1 2
ž
aq q b q 2
1rqy1
/
.
362
GH. TOADER
q f b b Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
= =
a sy1 2
as q b s
ž
2
2
2
ž
as q b s 2
1rsy1
/
1rsy1
q f b Ž Pq Ž a, b . , Ps Ž a, b . ; p, n .
/
a sy1 Ž 1 y s . b sy1 2
b sy1
ž
as q b s 2
1rsy2
/
.
Setting a s b s c yields f aa Ž c, c; p, n . y 2 f ab Ž c, c; p, n . q f b b Ž c, c; p, n . q
1yq c
f a Ž c, c; p, n . q
1ys c
f b Ž c, c; p, n . s 0.
Using Ži., Žii., and Žiii., we get cp0 Ž c . q Ž n y q y s q 1 . p9 Ž c . s 0, with solution Ž7.. We remark that the results are also valid if one of the parameters q and s vanishes.
3. POWER MEANS As was shown in w4, 5x, the means M p, n can represent some known means for special choices of p and n. In this section we want to determine the functions p for which M p, n is a power mean Pq , where n and q are arbitrary real numbers. THEOREM 2. If for some twice continuously differentiable function p the mean M p, n reduces at the power mean Pq , then p s a e2 qyn q b , where a and b are arbitrary real numbers. Proof. Using Ž4. and Ž5., we have M p , n Ž a, b . s Pq Ž a, b . if and only if f Ž a, b; p, n . s p Ž Pq Ž a, b . . .
Ž 79 .
363
SOME MEAN VALUES
Applying 2r a b to both sides, we get
f ab Ž a, b; p, n . s p0 Ž Pq Ž a, b . .
ž
aq q b q
q p9 Ž Pq Ž a, b . .
2
2 Ž1rqy1 .
/
b qy1 a qy1 2
a qy1 Ž 1 y q . b qy1 2
2
2
ž
aq q b q 2
1rqy2
/
.
Setting a s b s c and using the lemma yields cp0 Ž c . y Ž n y 1 . p9 Ž c . r2 c s p0 Ž c . q p9 Ž c . Ž 1 y q . rc. Thus cp0 Ž c . s Ž 2 q y n y 1 . p9 Ž c . , with the solution Ž79.. Remark. For n s 2 and q s 0, 1, and 2, the result was proved in w4x. For n s 1 and q s 0, 1, and 1r2 or n s y1 and q s y1, y1r2, and 0, it was proved in w5x. In all these cases the condition was proved to be also sufficient, that is, M p, n s Pq . But this is not generally valid. For example, in the case n s 2 and p s ey1 , we have M p, n s U and not P1r2 . Also M p, 2 / Py1 s H for p s e4 as was proved in w4x. Other cases in which M p, n / Pq for p s e2 qyn were given in w5x as we shall show in what follows. 4. A RECURRENCE FORMULA In what follows we use f Ž a, b; p, n. and M p, n only for p s e q , so that we denote them by f Ž a, b; q, n. and Mq, n , respectively. Also, as Mq , n Ž a, b . s f Ž a, b; q, n .
1rq
for q / 0
and M0, n Ž a, b . s exp f Ž a, b; 0, n . , we usually content ourselves with determining the function f for some values of q and n.
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GH. TOADER
In w4, 5x the following values are given: f Ž a, b; 0, 2 . s log 1
f Ž a, b; y2, 2 . s
ab
aqb 2 ,
a2 q b 2
f Ž a, b; 2, 2 . s
2 1
f Ž a, b; y2, 1 . s
ž
2'ab 1
f Ž a, b; y1, 1 . s
ž
, 1 a
q
1
/
b
,
Ž 9.
,
'ab
f Ž a, b; 0, 1 . s log
,
Ž 10 . 2
a1r2 q b1r2
/
2
.
As we have already mentioned, 2
f Ž a, b; y1, 2 . s
pa
K
ž(
1y
2
b
ž // a
,
Ž 11 .
where K is the complete elliptic integral of the first kind. Similarly, f Ž a, b; 1, 2 . s
2a
p
E
ž(
1y
b
2
ž // a
,
Ž 12 .
where E is the complete elliptic integral of the second kind. To obtain new formulas, let us begin by remarking that f Ž a, b; q, n . s f Ž a n , b n ; qrn, 1 . .
Ž 13 .
Thus, for the moment, it is enough to consider the case n s 1. As in Ž49. we have f Ž a, b; q, 1 . s s
2
p 2
p
pr2
H0
ž
aqb 2
q
Ž a cos 2u q b sin 2u . du q
/
pr2
H0
ž
1q
ayb aqb
q
cos 2 u
/
du .
365
SOME MEAN VALUES
If we denote Jq s
pr2
H0
q
Ž 1 q k cos 2 u . du
and
2
aqb
ks
ayb aqb
,
we get f Ž a, b; q, 1 . s
p
ž
2
q
/
Jq .
Ž 14 .
We look for a recurrence formula for Jq . We have Jq s
pr2
H0
Ž 1 q k cos 2 u . Ž 1 q k cos 2 u .
s Jqy 1 q s Jqy1 q q
pr2
H0
k
pr2
H 2 0 k 2
qy 1
Ž sin 2 u . 9 Ž 1 q k cos 2 u .
sin 2 u Ž 1 q k cos 2 u .
du qy 1
qy 1 p r2 <0
sin 2 2 u Ž q y 1 . k 2 Ž 1 q k cos 2 u .
s Jqy 1 q k 2 Ž q y 1 .
pr2
H0
du
qy 2
du
Ž 1 y cos 2 2 u . Ž 1 q k cos 2 u .
qy 2
du
s Jqy 1 q k 2 Ž q y 1 . Jqy2 y Ž q y 1 . =
pr2
H0
2 Ž k cos 2 u q 1 . y 2 Ž k cos 2 u q 1 . q 1
= Ž 1 y k cos 2 u .
qy 2
du
s Jqy 1 q k 2 Ž q y 1 . Jqy2 y Ž q y 1 . Ž Jq y 2 Jqy1 q Jqy2 . . Thus qJq s Ž 2 q y 1 . Jqy1 q Ž q y 1 . Ž k 2 y 1 . Jqy2 . Of course, q, q y 1, and q y 2 are different from 0. As k2 y 1 s
y4 ab
Ž a q b.
2
,
Ž 15 .
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GH. TOADER
the relations Ž14. and Ž15. give qf Ž a, b; q, 1 . y Ž q y 1r2 . Ž a q b . f Ž a, b; q y 1, 1 . q ab Ž q y 1 . f Ž a, b; q y 2, 1 . s 0.
Ž 16 .
Replacing a and b by a n and b n, respectively, and rewriting Ž13. as f Ž a n , b n ; q, 1 . s f Ž a, b; nq, n . ,
Ž 139.
we have qf Ž a, b; nq, n . y Ž q y 1r2 . Ž a n q b n . f Ž a, b; n Ž q y 1 . , n . q a n b n Ž q y 1 . f Ž a, b; n Ž q y 2 . , n . s 0. Using these relations, starting from two adequate values of q, we can obtain successively more values of f Ž a, b; q, n.. For example, using Ž9., Ž10., and Ž16., we get f Ž a, b; y3, 1 . s
3a2 q 2 ab q 3b 2 8 Ž ab .
5r2
.
Then f Ž a, b; y4, 1 . s
5a3 q 3a2 b q 3ab 2 q 5b 3 16 Ž ab .
7r2
and so on. Also it is easy to see that f Ž a, b; 1, 1 . s
aqb 2
and f Ž a, b; 2, 1 . s
3a2 q 2 ab q 3b 2 8
.
From Ž16. we have f Ž a, b; 3, 1 . s
5a3 q 3a2 b q 3ab 2 q 5b 3 16
.
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SOME MEAN VALUES
Then f Ž a, b; 4, 1 . s
35a4 q 20 a3 b q 18 a2 b 2 q 20 ab 3 q 35b 4 128
and so on. Using Ž13., we can rewrite Ž11. and Ž12. as f a, b; y
ž
1 2
, 1 s f Ž 'a , 'b ; y1, 2 . s
/
2
K
p'a
b
ž( / 1y
a
and
ž
f a, b;
1 2
2'a
, 1 s f Ž 'a , 'b ; 1, 2 . s
/
p
E
b
ž( / 1y
a
.
Taking in Ž16. q s 3r2, we get
ž
f a, b;
3 2
,1 s
/
2'a 3p
2Ž a q b . E
b
b
ž( / ž( / 1y
y bK
a
1y
a
.
Then, for q s 1r2, f a, b; y
ž
3 2
2
,1 s
/
p b'a
E
b
ž( / 1y
a
and so on, we can express any f Ž a, b; Ž2 q q 1.r2, 1. by means of K and E. In fact, for n s 1, we are able to determine any function f Ž a, b; q, 1. for q s k or q s Ž2 k q 1.r2, with k integer, but only for such values of q. All these functions can be transcribed by Ž139. to give their expressions for an arbitrary n / 0: f Ž a, b; nq, n . s f Ž a n , b n ; q, 1 . . We reobtain all the means given in w4, 5x, but we can also determine many other means Mq, n . For n s 0, it is easy to prove that M0, 0 s G, but for q / 0 we do not know how to determine any mean Mq , 0 Ž a, b . s
ž
1
2p q cos 2 u
H 2p 0
a
1rq
b q sin
2
u
du
/
.
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GH. TOADER
REFERENCES 1. J. Aczel, ´ The notion of mean value, Norske Vid. Selsk. Forh. ŽTrondheim. 19 Ž1946., 83]86. 2. P. S. Bullen, D. S. Mitrinovic, ´ and P. M. Vasic, ´ ‘‘Means and Their Inequalities,’’ Reidel, Dordrecht, 1988. 3. C. Gini, ‘‘Means,’’ Unione Tipografico-Editrice Torinese, Milan, 1958. wIn Italian x 4. H. Haruki, New characterizations of the arithmetic]geometric mean of Gauss and other well-known mean values, Publ. Math. Debrecen 38 Ž1991., 323]332. 5. H. Haruki and T. M. Rassias, New characterizations of some mean-values, J. Math. Anal. Appl. 202 Ž1996., 333]348. 6. M. K. Vamanamurthy and M. Vuorinen, Inequalities for means, J. Math. Anal. Appl. 183 Ž1994., 155]166.