Asymptotical formulas for Gaussian and generalized hypergeometric functions

Applied Mathematics and Computation 276 (2016) 44–60

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Asymptotical formulas for Gaussian and generalized hypergeometric functions Miao-Kun Wang a, Yu-Ming Chu a,∗, Ying-Qing Song b a b

Department of Mathematics, Huzhou University, Huzhou 313000, China School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

a r t i c l e

i n f o

a b s t r a c t

MSC: 33C05 33A30

In this paper, we present several generalizations and refinements for the asymptotic formulas of Gaussian and generalized hypergeometric functions. © 2015 Elsevier Inc. All rights reserved.

Keywords: Gaussian hypergeometric function Generalized hypergeometric function Asymptotical formula Monotonicity Inequality

1. Introduction 1.1. Gaussian hypergeometric function 2 F1 For real numbers a, b, and c with c = 0, −1, −2, . . . , the Gaussian hypergeometric function 2 F1 is defined by 2 F1

(a, b; c; x ) = F (a, b; c; x ) =

∞  ( a ) n ( b ) n xn (c )n n! n=0

for |x| < 1, where the Pochhammer symbol

( a )n = a ( a + 1 ) · · · ( a + n − 1 ) =

(a + n ) , (a )

∞ for n = 1, 2, . . . , and (a )0 = 1 for a = 0, (x ) = 0 t x−1 e−t dt (x > 0) is the gamma function. It is well known that the Gaussian hypergeometric function has many important applications in geometric function theory, number theory and several other contexts, and a lot of special functions and elementary functions are the particular cases or limiting cases. Especially, in 1980s, de Branges used hypergeometric functions to prove the famous Bieberbach conjecture, which has given function theorists a renewed interest to study the role of Gaussian hypergeometric function. For the above, and more properties of F(a, b; c; x), see [1,3,5–8,10,11,13,16,19,21,24,25].

∗

Corresponding author. Tel.: +86 572 2321510; fax: +86 572 2321163. E-mail addresses: [email protected] (M.-K. Wang), [email protected], [email protected] (Y.-M. Chu), [email protected] (Y.-Q. Song).

http://dx.doi.org/10.1016/j.amc.2015.11.088 0096-3003/© 2015 Elsevier Inc. All rights reserved.

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

45

The function F(a, b; c; x) is said to be zero-balanced if c = a + b. In the zero-balanced case, there is a logarithmic singularity at x = 1, and Gauss proved the asymptotic formula (see [14])

F (a, b; a + b; x ) ∼ −

1 log(1 − x ), B(a, b)

x → 1,

(1.1)

where

B(z, w ) =

(z )(w ) , (z + w )

(z ) > 0,

 (w ) > 0

(1.2)

is the classical Beta function. Ramanujan found a much sharper asymptotic formula (see [14])

B(a, b)F (a, b; a + b; x ) + log(1 − x ) = R(a, b) + O((1 − x ) log(1 − x )),

x → 1,

(1.3)

where

R(a, b) = −ψ (a ) − ψ (b) − 2γ ,

(1.4)

ψ (z ) =   (z )/(z ), (z) > 0 and γ is the Euler–Mascheroni constant. In order to refine Gauss’ asymptotic formula (1.1), Anderson et al. [4] considered the following Problem 1.1. Problem 1.1. Is it true that the function

x → F (a, b; a + b; x ) +

1 1 log(1 − x ), B(a, b) x

is monotone on (0, 1) for suitable a and b? The above function was shown to be monotone for a, b ∈ (0, 1) or (a, b) ∈ (1, ∞) in [2]. Indeed, they proved that Theorem 1.1 ([ [2], Theorem 1.4]). Let a, b ∈ (0, ∞), B = B(a, b), R = R(a, b), x ∈ (0, 1) and

f (x ) =

xF (a, b; a + b; x ) . log[1/(1 − x )]

Then the following statement are true: (1) (2) (3) (4) (5)

f is decreasing from (0, 1) onto (1/B, 1) if a, b ∈ (0, 1); f is increasing from (0, 1) onto (1, 1/B) if a, b ∈ (1, ∞); f(x) ≡ 1 for all x ∈ (0, 1) if a = b = 1; The function g(x ) = BF (a, b; a + b; x ) + (1/x ) log(1 − x ) is increasing from (0, 1) onto (B − 1, R ) if a, b ∈ (0, 1); The function g(x ) = BF (a, b; a + b; x ) + (1/x ) log(1 − x ) is decreasing from (0, 1) onto (R, B − 1 ) if a, b ∈ (1, ∞).

Later, Qiu and Vuorinen [20] proved the following Theorems 1.2 and 1.3, where Theorem 1.2 extended the parts (4) and (5) in Theorem 1.1, while Theorem 1.3 is another refinement of Gauss’ asymptotic formula (1.1). Theorem 1.2 ([ [20], Theorem 1.4]). Let a, b ∈ (0, ∞), c = a + b, a∗1 = 1 − ab, a∗2 = 2ab − a − b, a∗3 = |a∗1 | + |a∗2 |, B = B(a, b), R = R(a, b), x ∈ (0, 1)

g(x ) = BF (a, b; c; x ) +

1 log(1 − x ). x

Then we have (1) (2) (3) (4)

g(x) ≡ 0 for all x ∈ (0, 1) if a∗3 = 0; g is strictly increasing from (0, 1) onto (B − 1, R ) if a∗3 = 0 and a∗1 ≥ max{0, a∗2 }; g is strictly decreasing from (0, 1) onto (R, B − 1 ) if a∗3 = 0 and a∗1 ≤ min{0, a∗2 }; In the other case not stated in parts (1)-(3), that is a∗2 < a∗1 < 0, g is not always monotone on (0, 1).

Theorem 1.3 ([ [20], Theorem 1.5]). Let a, b ∈ (0, ∞), A∗1 = A∗1 (a, b) = a + b + ab − 3, A∗2 = A∗2 (a, b) = a + b − 3ab + 1, A∗ = |A∗1 | + |A∗2 |, B = B(a, b), R = R(a, b), r ∈ (0, 1) and

h (r ) =

BF (a, b; a + b; r ) + log(1 − r ) − R . [(1 − r )/r] log[1/(1 − r )]

Then we have (1) h(r) ≡ 1 for all r ∈ (0, 1) if A∗ = 0; (2) h is strictly decreasing from (0, 1) onto (ab, B − R ) if A∗ = 0 and A∗1 ≤ min{0, A∗2 }. In particular, with this condition, for all r ∈ (0, 1),



ab

1−r 1 log r 1−r



< BF (a, b; a + b; r ) + log(1 − r ) − R < (B − R )





1−r 1 log ; r 1−r

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M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

(3) h is strictly increasing from (0, 1) onto (B − R, ab) if A∗ = 0 and A∗1 ≥ max{0, A∗2 }. In particular, with this condition, for all r ∈ (0, 1),

(B − R )



1−r 1 log r 1−r



< BF (a, b; a + b; r ) + log(1 − r ) − R < ab





1−r 1 log ; r 1−r

(4) In the other cases not stated in parts (1)–(3), namely, 0 < A∗1 < A∗2 , h is not always monotone on (0, 1). 1.2. Generalized hypergeometric function

p+1 Fp

For a generalization of the Gaussian summation formula, the generalized hypergeometric series or function with more parameters, p+1 Fp , is defined by



p+1 Fp

The function

s=

a1 , a2 , . . . , a p+1 b1 , b2 , . . . , b p

p+1 Fp

p 

bj −

j=1

   ∞ (a1 )n (a2 )n . . . (a p+1 )n xn  , |x| < 1. x = (b ) . . . (b ) n! n=0

1 n

(1.5)

p n

is called s-balanced (zero-balanced) if the parameter difference p+1 

aj

(1.6)

j=1

is equal to an integer (zero). In the past few years, the zero-balanced generalized hypergeometric function has received particular attention, especially their value at unit argument. For example, when p = 2, the asymptotical formula of the zero-balanced generalized hypergeometric function,

   (a1 )(a2 )(a3 ) a1 , a2 , a3  x = − log(1 − x ) + L + O((1 − x ) log(1 − x )), 3 F2 b1 , b2 (b1 )(b2 )

x → 1,

if (a3 ) > 0, where

L = 2(1 ) − 2(a1 ) − (a2 ) +

∞  (b2 − a3 )k (b1 − a3 )k , k(a1 )k (a2 )k k=1

which is given by Ramanujan in [22](without proof), was completely proved by Berndt [9], and by Evans and Stanton [15]. Later Saigo and Srivastava [23] found the leading terms of the behavior when x → 1 for arbitrary p in zero-balanced case. Bühring [12] derived the analytic continuation formulas of p+1 Fp for arbitrary p and unrestricted s. Corollary 1.1 ([ [12], Corollary 1]). If s according to (1.6) is equal to zero, then the continuation formula

   (a1 )(a2 ) . . . (a p+1 ) a1 , a2 , . . . , a p+1  x p+1 Fp b1 , b2 , . . . , b p (b1 ) . . . (b p ) ∞ n  (a1 )n (a2 )n  (−n )k = A( p) {(1 + n − k ) + (1 + n ) − (a1 + n ) − (a2 + n ) − log(1 − x )} n!n! ( a1 )k (a2 )k k n=0 k=0

∞  ( k − n − 1 )! ( p ) n + (−1 ) n! A ( 1 − x )n (a1 )k (a2 )k k

(1.7)

k=n+1

holds for |1 − x| < 1, | arg(1 − x )| < π , and p = 2, 3, . . . , where the condition (a3 + n ) > 0 ∧ · · · ∧ (a p+1 + n ) > 0 is required for ( p)

convergence of the infinite series, and the Ak

Ak(2) =

Ak(3) =

(b2 − a3 )k (b1 − a3 )k k!

are given as follows

,

k  (b3 + b2 − a4 − a3 + k2 )k−k2 (b1 − a3 )k−k2 (b3 − a4 )k2 (b2 − a4 )k2 , (k − k2 )!k2 !

k2 =0

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

Ak(4) =

47

k  (b4 + b3 + b2 − a5 − a4 − a3 + k2 )k−k2 (b1 − a3 )k−k2 (k2 − k3 )!

k2 =0

×

k2  (b4 + b3 − a5 − a4 + k3 )k2 −k3 (b2 − a4 )k2 −k3 (b4 − a5 )k3 (b3 − a5 )k3 , (k2 − k3 )! k3 !

k3 =0

and

Ak( p) =

k  (b p + b p−1 + · · · + b2 − a p+1 − a p − · · · − a3 + k2 )k−k2 (b1 − a3 )k−k2 (k − k2 )!

k2 =0

×

k2  (b p + b p−1 + · · · + b3 − a p+1 − a p − · · · − a4 + k3 )k2 −k3 (b2 − a4 )k2 −k3 ··· (k2 − k3 )!

k3 =0

×

k p−2  (b p + b p−1 − a p+1 − a p + k p−1 )k p−2 −k p−1 (b p−2 − a p )k p−2 −k p−1 (b p − a p+1 )k p−1 (b p−1 − a p+1 )k p−1

(k p−2 − k p−1 )!

k p−1 =0

k p−1 !

.

So, when a1 , a2 , . . . , a p+1 , b1 , b2 , . . . , b p ∈ R and x ∈ (−1, 1 ), the behavior when x → 1 of the zero-balanced series is

(a1 )(a2 ) . . . (a p+1 ) (b1 ) . . . (b p )



p+1 Fp

a1 , a2 , . . . , a p+1 b1 , b2 , . . . , b p

   x

= [R(a1 , a2 ) + M][1 + O(1 − x )] − log(1 − x )[1 + O(1 − x )],

(1.8)

x → 1,

where

M=

∞  ( k − 1 )! ( p ) A (a1 )k (a2 )k k

(1.9)

k=1

and R(·, ·) is defined as in (1.4). Problem 1.2 ([ [2], 4.10 Conjectures(2)]). Find a generalization of Theorem 1.1 to the generalized hypergeometric function  p q a ,a ,...,a p  x ) when p = q + 1, ai > 0, bj > 0, i=1 ai = j=1 b j . p Fq (b1 ,b2 ,...,b 1

2

q

The main purpose of this paper is to refine Theorems 1.1–1.3 and give a generalization of Theorems 1.1 for the zero-balanced generalized hypergeometric function p+1 Fp . Our main results extend the asymptotic formulas (1.1) and (1.8). 2. Main results For convenience, we introduce some regions in {(a, b) ∈ R2 |a > 0, b > 0}:(see Appendix: Figs. 1–4)

D1 = {(a, b)|a, b > 0, ab ≤ 1, 2ab − (a + b) ≤ 0} = {(a, b)|a, b > 0, ab ≤ 1}, D2 = {(a, b)|a, b > 0, ab > 1, R(a, b) ≥ 0}, D3 = {(a, b)|a, b > 0, ab ≥ 1, 2ab − (a + b) ≥ 0} = {(a, b)|a, b > 0, 2ab − (a + b) ≥ 0}, D4 = {(a, b)|a, b > 0, R(a, b) < 0, 2ab − (a + b) < 0}, E1 = {(a, b)|a, b > 0, a + b + ab − 3 ≤ 0}, E2 = {(a, b)|a, b > 0, a + b + ab − 3 > 0, ab ≤ 1}, E3 = {(a, b)|a, b > 0, 3ab − (a + b + 1 ) ≥ 0}, E4 = {(a, b)|a, b > 0, ab > 1, 3ab − (a + b + 1 ) < 0}, E41 = {(a, b)|a, b > 0, 2abB(a, b) > a + b, ab > 1}, E42 = {(a, b)|a, b > 0, 3ab − (a + b + 1 ) < 0, 2abB(a, b) ≤ a + b},

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M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

Fig. 1. The regions D1 , D2 , D3 , D4 .

1 = {(a, b)|a, b > 0, a + b + ab − 3 ≤ 0} = E1 , 2 = {(a, b)|a, b > 0, a + b + ab − 3 > 0, R(a + 1, b + 1 ) ≥ −2}, 3 = {(a, b)|a, b > 0, ab ≥ 1}, 4 = {(a, b)|a, b > 0, ab < 1, R(a + 1, b + 1 ) < −2}, 41 = {(a, b)|a, b > 0, ab < 1, (a + b)(a + b + 1 ) − abB(a, b)(3ab + a + b + 1 ) ≥ 0}, 42 = {(a, b)|a, b > 0, (a + b)(a + b + 1 ) − abB(a, b)(3ab + a + b + 1 ) < 0, R(a + 1, b + 1 ) < −2}, ∗42 = {(a, b)|a, b > 0, (a + b)(a + b + 1 ) − abB(a, b)(3ab + a + b + 1 ) < 0, [R(a, b) + 2 − B(a, b)](a + b) − 2abB(a, b) ≤ 0}, ∗∗ 42 = { (a, b)|a, b > 0, [R (a, b) + 2 − B (a, b)] (a + b) − 2abB (a, b) > 0, R (a + 1, b + 1 ) < −2},  1 = 1 ∪ 2 ,

3 = 3 ∪ 41 ∪ ∗42 ,

Clearly, D1 ∪ D2 ∪ D3 ∪ D4 = E1 ∪ E2 ∪ E3 ∪ E4 = 1 ∪ 2 ∪ 3 ∪ 4 = {(a, b) ∈ R2 |a > 0, b > 0}, E4 = E41 ∪ E42 , 4 = 41 ∪ 42 , 42 = ∗42 ∪ ∗∗ . 42 We now state our main results. Theorem 2.1. For a, b ∈ (0, ∞), let

f (x ) =

xF (a, b; a + b; x ) log[1/(1 − x )]

on (0, 1). Then we have

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

49

Fig. 2. The regions E1 , E2 , E3 , E4 , E41 , E42 .

(1) (2) (3) (4)

If a = b = 1, then f(x) ≡ 1 for all x ∈ (0, 1); If (a, b) = (1, 1) and (a, b) ∈ D1 ∪ D2 , then f is strictly decreasing with range (1/B, 1), where B = B(a, b); If (a, b) = (1, 1) and (a, b) ∈ D3 , then f is strictly increasing with range (1, 1/B); If (a, b) = (1, 1) and (a, b) ∈ D4 , then there exists x0 ∈ (0, 1) such that f is strictly decreasing in (0, x0 ), and strictly increasing in (x0 , 1).

Theorem 2.2. For a, b ∈ (0, ∞), let

gc (x ) = F (a, b; a + b; x ) +

c log(1 − x ), x

c∈R

on (0, 1). Then we have (1) When a = b = 1, gc is strictly decreasing (increasing) on (0, 1) if c > 1 (c < 1), and gc (x ) = 0 for all x ∈ (0, 1) if c = 1/B(a, b) = 1; (2) When (a, b) = (1, 1) and (a, b) ∈ E1 ∪ E2 , gc is strictly increasing if c ≤ 1/B(a, b), and strictly decreasing if c ≥ 2ab/(a + b), furthermore, if 1/B(a, b) < c < 2ab/(a + b), then there exists x∗0 ∈ (0, 1 ) such that gc is strictly increasing in (0, x∗0 ), and strictly decreasing in (x∗0 , 1 ); (3) When (a, b) = (1, 1) and (a, b) ∈ E3 , gc is strictly increasing if c ≤ 2ab/(a + b), and strictly decreasing if c ≥ 1/B(a, b), furthermore, if 2ab/(a + b) < c < 1/B(a, b), then there exists x1 ∈ (0, 1) such that gc is strictly decreasing in (0, x1 ), and strictly increasing in (x1 , 1); (4) When (a, b) = (1, 1) and (a, b) ∈ E41 , gc is strictly decreasing if c ≤ 2ab/(a + b), and if c = 1/B(a, b), then there exists x2 ∈ (0, 1) such that gc is strictly increasing in (0, x2 ), and strictly decreasing in (x2 , 1); (5) When (a, b) = (1, 1) and (a, b) ∈ E42 , gc is strictly decreasing if c ≤ 1/B(a, b), and if c = 2ab/(a + b), then there exists x3 ∈ (0, 1) such that gc is strictly decreasing in (0, x3 ), and strictly increasing in (x3 , 1). Corollary 2.1. For a, b ∈ (0, ∞), let

g(x ) = B(a, b)F (a, b; a + b; x ) + on (0, 1). Then we have

1 log (1 − x ) x

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M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

Fig. 3. The regions 1 , 2 , 3 , 4 , 41 , 42 , ∗42 , ∗∗ 42 .

(1) (2) (3) (4)

If a = b = 1, then g(x) ≡ 0 for all x ∈ (0, 1); If (a, b) = (1, 1) and (a, b) ∈ E1 ∪ E2 , then g is strictly increasing from (0, 1) onto (B − 1, R ), where B = B(a, b) and R = R(a, b); If (a, b) = (1, 1) and (a, b) ∈ E3 ∪ E42 , then g is strictly decreasing from (0, 1) onto (R, B − 1 ); If (a, b) = (1, 1) and (a, b) ∈ E41 , then g is strictly increasing in (0, x2 ), and strictly decreasing in (x2 , 1).

Remark 2.1. It is not difficult to find that, in Theorem 1.2, g is strictly increasing in (0, 1) for (a, b) ∈ E1 ∪ E2 \{(1, 1)}, and strictly decreasing for (a, b) ∈ E3 \{(1, 1)}, while our Theorem 2.2 gives that g is also strictly decreasing in (0, 1) for (a, b) ∈ E42 \{(1, 1)}, and when (a, b) ∈ E41 \{(1, 1)}, g is piecewise monotone on (0, 1). Theorem 2.3. For a, b ∈ (0, ∞), define the function f on (0, 1) by

h (r ) =

BF (a, b; a + b; r ) + log(1 − r ) − R , [(1 − r )/r] log[1/(1 − r )]

where B = B(a, b) and R = R(a, b) are defined by (1.2) and (1.4), respectively. Then we have the following conclusions: (1) If a = b = 1, then h(r) ≡ 1; (2) If (a, b) = (1, 1) and (a, b) ∈  1 ≡ 1 ∪ 2 , then h is strictly decreasing from (0, 1) onto (ab, B − R ). In particular, with this condition, for all r ∈ (0, 1),



ab

1−r 1 log r 1−r



< BF (a, b; a + b; r ) + log(1 − r ) − R < (B − R )





1−r 1 log ; r 1−r

(2.1)

(3) If (a, b) = (1, 1) and (a, b) ∈ 3 ≡ 3 ∪ 41 ∪ ∗42 , then h is strictly increasing from (0, 1) onto (B − R, ab). In particular, with this condition, for all r ∈ (0, 1),

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

51

Fig. 4. The regions  1 ,  3 , ∗∗ 2 .

(B − R )



1−r 1 log r 1−r



< BF (a, b; a + b; r ) + log(1 − r ) − R < ab





1−r 1 log ; r 1−r

(2.2)

(4) In the remaining region ∗∗ 42 , there exists λ ∈ (0, 1) such that h is strictly decreasing in (0, λ), and strictly increasing in (λ, 1). Consequently,

BF (a, b; a + b; r ) + log(1 − r ) − R < max{B − R, ab}





1−r 1 log . r 1−r

(2.3)

Remark 2.2. By simple computation, we clearly see that, in Theorem 1.3, h is strictly decreasing in (0, 1) for (a, b) ∈ 1 \{(1, 1)}, and strictly increasing for (a, b) ∈ 3 \{(1, 1)}, while our Theorem 2.3 gives a complete answer to the monotonicity of h on (0, 1) for arbitrary (a, b) ∈ {(a, b)|a > 0, b > 0}. Theorem 2.4. For a1 , a2 , . . . , a p+1 , b1 , . . . , b p ∈ (0, ∞ ) with a1 + a2 + · · · + a p+1 = b p + b p−1 + · · · + b1 , let

1 ( 2 ) = {(a1 , a2 , . . . , a p+1 , b1 , . . . , b p )|a1 + a2 + · · · + a p+1 = b p + b p−1 + · · · + b1 , 

ai1 ai2 . . . aik −

1≤i1


ai1 ai2 . . . aik−1 −

1≤i1
−



1≤i1
b i1 b i2 . . . b ik

1≤i1


+2



bi1 bi2 . . . bik−2 ≤ (≥ )0,



bi1 bi2 . . . bik−1

1≤i1
k = 2, 3, . . . , p + 2}

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M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

and

B=

(a1 )(a2 ) . . . (a p+1 ) . (b1 ) . . . (b p )

Define the function F on (0, 1) by



x

p+1 Fp

F (x ) =

a1 , a2 , . . . , a p+1 b1 , b2 , . . . , b p

log[1/(1 − x )]

   x .

Then we have (1) If (a1 , a2 , . . . , a p+1 , b1 , . . . , b p ) ∈ 1 , then F is decreasing with range (1/B, 1 ); (2) If (a1 , a2 , . . . , a p+1 , b1 , . . . , b p ) ∈ 2 , then F is increasing with range (1, 1/B ). Remark 2.3. When p = 1, the regions 1 and 2 reduce to

D1 = {a1 , a2 > 0|a1 a2 ≤ 1, 2a1 a2 − (a1 + a2 ) ≤ 0}, D3 = {a1 , a2 > 0|a1 a2 ≥ 1, 2a1 a2 − (a1 + a2 ) ≥ 0}, respectively. Since (0, 1) × (0, 1) ⊂ D1 , and (1, ∞) × (1, ∞) ⊂ D3 , Our Theorem 2.4 is a generalization of Theorem 1.1(1) and (2). Theorem 2.5. For a1 , a2 , . . . , a p+1 , b1 , . . . , b p ∈ (0, ∞ ) with a1 + a2 + · · · + a p+1 = b p + b p−1 + · · · + b1 , let

3 ( 4 ) = {(a1 , a2 , . . . , a p+1 , b1 , . . . , b p )|a1 + a2 + · · · + a p+1 = b p + b p−1 + · · · + b1 , 

1≤i1




(ai1 + 1 ) . . . (aik−1 + 1 ) − 4

1≤i1
−4

(bi1 + 1 )(bi2 + 1 ) . . . (bik + 1 )

1≤i1


+3



(ai1 + 1 )(ai2 + 1 ) . . . (aik + 1 ) −

(bi1 + 1 ) . . . (bik−1 + 1 )

1≤i1
(bi1 + 1 ) . . . (bik−2 + 1 ) ≤ (≥ )0,

k = 2, 3, . . . , p + 2}

1≤i1
and B be defined as in Theorem 2.4. Define the function G on (0, 1) by



G (x ) = B

p+1 Fp

a1 , a2 , . . . , a p+1 b1 , b2 , . . . , b p

  1  x + x log(1 − x ).

(2.4)

Then we have (1) If (a1 , a2 , . . . , a p+1 , b1 , . . . , b p ) ∈ 3 , then G is increasing with range (B − 1, R(a1 , a2 ) + M ); (2) If (a1 , a2 , . . . , a p+1 , b1 , . . . , b p ) ∈ 4 , then G is increasing with range (R(a1 , a2 ) + M, B − 1 ). Remark 2.4. When p = 1, the regions 3 and 4 reduce to

E1 ∪ E2 = {a1 , a2 > 0|a1 a2 ≤ 1, 3a1 a2 − (a1 + a2 + 1 ) ≤ 0} = {a1 , a2 > 0|a1 a2 ≤ 1}, E3 = {a1 , a2 > 0|a1 a2 ≥ 1, 3a1 a2 − (a1 + a2 + 1 ) ≥ 0} = {a1 , a2 > 0|3a1 a2 − (a1 + a2 + 1 ) ≥ 0}, respectively. Since (0, 1) × (0, 1) ⊂ E1 ∪ E2 , and (1, ∞) × (1, ∞) ⊂ E3 , Our Theorem 2.5 is also a generalization of Theorem 1.1(3) and (4). 3. Proofs of theorems At the beginning of this section, we shall introduce an important auxiliary function and two technical lemmas used afterward. Let −∞ ≤ a < b ≤ ∞, f and g be differentiable on (a, b), and g = 0 on (a, b). Then the function Hf, g is defined by

f H f,g ≡  g − f. g Lemma 3.1 (see [[18], Lemma 2.1], [[26], Theorem 2.1]). Let A(t ) = verging on (−r, r ) and bk > 0 for all k.

(3.1) ∞

k=0

ak t k and B(t ) =

∞

k=0

bk t k be two real power series con-

(i) Suppose that the non-constant sequence {ak /bk }∞ is increasing (decreasing), then A/B is strictly increasing (decreasing) on (0, k=0 r);

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

53

(ii) Suppose that for certain m ∈ N, the non-constant sequence {ak /bk } is increasing (decreasing) for 0 ≤ k ≤ m and decreasing (increasing) for k ≥ m. Then the function A/B is strictly increasing (decreasing) on (0, r) if and only if HA,B (r− ) ≥ (≤ )0. Moreover, if HA,B (r− ) < (> )0, then there exists t0 ∈ (0, r) such that the function A/B is strictly increasing (decreasing) on (0, t0 ) and strictly decreasing (increasing) on (t0 , r). Lemma 3.2 (see [[5], Theorem 1.25], [[17], Proposition 4.4]). For −∞ ≤ a < b ≤ ∞, let f and g be differentiable functions on (a, b) with g = 0 on (a, b), and f (b− ) = g(b− ) = 0. Then we have (i) If f /g is (strictly) increasing (decreasing) on (a, b), then so is f/g; (ii) Suppose that there exists c ∈ (a, b) such that f /g is strictly increasing (decreasing) on (a, c) and strictly decreasing (increasing) on (c, b). Then if sgng sgnH f,g (a+ ) ≤ (≥ )0, then f/g is strictly decreasing (increasing) on (a, b), and if sgng sgnH f,g (a+ ) > (< )0, then there is a unique number xb ∈ (a, b) such that f/g is strictly increasing (decreasing) on (a, xb ) and strictly decreasing (increasing) on (xb , b). Here Hf, g is defined on (a, b) by (3.1), and sgn(·) is the sign function. Lemma 3.3. Define the functions F, G, F∗ , G∗ on (0, 1) by

F (x ) = F (a, b; a + b; x ),

G(x ) = F (a, b; a + b + 1; x ),

and

F ∗ (x ) = F (a1 , b1 ; a1 + b1 ; x ),

G∗ (x ) = F (a1 , b1 ; a1 + b1 + 1; x ).

where a, b, a1 , b1 > 0 with (a, b) = (a1 , b1 ) and (a, b) = (b1 , a1 ). Then one has

lim HF,F ∗ (x ) =

x→1−

lim− HG,G∗ (x ) =

x→1

1 [R(a1 , b1 ) − R(a, b)], B(a, b)

(3.2)

  (a + b + 1 ) ab −1 . (a + 1 )(b + 1 ) a1 b1

(3.3)

Proof. Since (see [[5], Theorem 1.19(4) and (10)])

(1 − x )F  (a, b; a + b; x ) = (1 − x ) =

F (a, b; c; 1 ) =

ab F (a + 1, b + 1; a + b + 1; x ) a+b

ab F (a, b; a + b + 1; x ), a+b

(c − a − b)(c ) , (c − a )(c − b)

c > a + b,

(3.4)

we have

G(1− ) = F (a, b; a + b + 1; 1 ) = G∗ ( 1 − ) =

(a1 + b1 + 1 )(1 ) (a1 + 1 )(b1 + 1 )

(a + b + 1 )(1 ) , (a + 1 )(b + 1 )

and

lim−

x→1

F  (x )

( 1 − x )F  ( x ) ab a1 + b1 G(x ) = lim− x→1 (1 − x )F ∗  (x ) x→1 a + b a1 b1 G∗ (x ) F (x ) (a + b) B(a1 , b1 ) (a )(b) = lim− = . x→1 (a1 + b1 ) B(a, b) (a1 )(b1 ) ∗

= lim−

Thus it follows from (1.3) that

lim− HF,F ∗ (x ) = lim−

x→1

x→1

F  (x )

F ∗  (x )



F ∗ (x ) − F (x ) = lim− x→1

log(1 − x ) B(a1 , b1 ) R(a1 , b1 ) − B(a, b) B(a1 , b1 ) B(a1 , b1 )

1 − [R(a, b) − log(1 − x )] + O((1 − x ) log(1 − x )) B(a, b) 1 = [R(a1 , b1 ) − R(a, b)], B(a, b) which implies (3.2).



54

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

Similarly, for (3.3), we have

lim−

x→1

G ( x ) ∗

G (x )

= lim− x→1

(

)

ab F a + 1, b + 1; a + b + 2; x a+b+1 a1 b1 F a1 + 1, b1 + 1; a1 + b1 + 2; x a1 +b1 +1

(

ab a1 + b1 + 1 = lim− x→1 a + b + 1 a1 b1 =



)



1 1 log 1−x B(a+1,b+1 ) 1 1 log 1−x B(a1 +1,b1 +1 )



+ O (1 )



+ O (1 )

ab(a1 + b1 + 1 ) B(a1 + 1, b1 + 1 ) , a1 b1 (a + b + 1 ) B(a + 1, b + 1 )

lim HG,G∗ (x ) = lim−

G ( x )

G∗ ( x ) − G ( x ) G∗  ( x ) ab(a1 + b1 + 1 ) B(a1 + 1, b1 + 1 ) (a1 + b1 + 1 ) (a + b + 1 ) = − a1 b1 (a + b + 1 ) B(a + 1, b + 1 ) (a1 + 1 )(b1 + 1 ) (a + 1 )(b + 1 )

x→1−

x→1

  (a + b + 1 ) ab = −1 . (a + 1 )(b + 1 ) a1 b1



Proof of Theorem 2.1. Let A(x ) = F (a, b; a + b; x ) and B(x ) = log[1/(1 − x )]/x, then

f (x ) =

∞ A n xn A (x ) = i=0 , ∞ n B (x ) i=0 Bn x

where

An =

( a )n ( b )n , (a + b)n n!

Bn =

( 1 )n ( 1 )n . (2 )n n!

Clearly, if a = b = 1, then f(x) ≡ 1. When (a, b) = (1, 1), it is not difficult to verify that the monotonicity of An /Bn depends on the sign of

Hn = (ab − 1 )n + 2ab − (a + b).

(3.5)

On the other hand, by Lemma 3.3, the limiting value HA, B (x) at x = 1 is

lim HA,B (x ) = −

x→1−

R(a, b) . B(a, b)

(3.6)

Next, we divide the proof into three cases. Case 1 (a, b) ∈ D1 ࢨ{(1, 1)}. Then it follows from (3.5) that Hn < 0 for n = 1, 2, . . .. Thus from Lemma 3.1(1) we conclude that f is strictly decreasing on (0, 1). Case 2 (a, b) ∈ D3 ࢨ{(1, 1)}. Similarly, by (3.5) we have that Hn > 0 for n = 1, 2, . . . , and consequently f is strictly increasing on (0, 1). Case 3 (a, b) ∈ D2 ∪ D4 ࢨ{(1, 1)}. Then (3.5) implies that the sequence An /Bn is decreasing and then increasing. It follows from Lemma 3.1(2) and Eq. (3.6) that f is strictly decreasing if and only if (a, b) ∈ D2 ࢨ{(1, 1)}, and if (a, b) ∈ D4 ࢨ{(1, 1)}, then there exists x0 ∈ (0, 1) such that f is strictly decreasing on (0, x0 ), and strictly increasing on (x0 , 1).  Lemma 3.4. For a, b ∈ (0, ∞), let B = B(a, b) and

ξ (x ) =

2ab F (a, b; a + b + 1; x ) , a+b F (1, 1; 3; x )

x ∈ (0, 1 ).

Then we have (1) (2) (3) (4)

If a = b = 1, then ξ (x) ≡ 1 for all x ∈ (0, 1); If (a, b) = (1, 1) and (a, b) ∈ E1 ∪ E2 , then ξ is strictly decreasing with range (1/B, 2ab/(a + b)); If (a, b) = (1, 1) and (a, b) ∈ E3 , then ξ is strictly increasing with range (2ab/(a + b), 1/B ); If (a, b) = (1, 1) and (a, b) ∈ E4 , then there exists δ ∈ (0, 1) such that ξ is strictly decreasing in (0, δ ), and strictly increasing in (δ , 1). In particular, if (a, b) ∈ E41 (E42 ), then the maximum of ξ on (0, 1) is 2ab/(a + b)(1/B ).

Proof. Clearly, ξ (0+ ) = 2ab/(a + b), (1 )n (1 )n /[(3 )n n!], then

x→

and

by

(3.4),

ξ (1− ) = 1/B(a, b).

∞ ∗ n An x F (a, b; a + b + 1; x ) A∗ ( x ) = n=0 ≡ ∗ . ∞ ∗ n F (1, 1; 3; x ) B (x ) n=0 Bn x

Let

A∗n = (a )n (b)n /[(a + b + 1 )n n!],

B∗n =

(3.7)

Simple computations show that the monotonicity of A∗n /B∗n depends on the sign of

Hn∗ = (a + b + ab − 3 )n + 3ab − (a + b + 1 ).

(3.8)

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

55

On the other hand, by Lemma 3.3, the limiting value HA∗ ,B∗ (x ) at x = 1 is

lim HA∗ ,B∗ (x ) =

x→1−

(a + b + 1 ) (ab − 1 ). (a + 1 )(b + 1 )

(3.9)

Therefore, it follows from (3.7)–(3.9) and Lemma 3.1 together with the similar argument in the proof of Theorem 2.1 that the assertions of Lemma 3.4 hold true.  Proof of Theorem 2.2. Clearly if a = b = 1, then gc (x ) = (1 − c )F (1, 1; 2; x ), and thereby the assertions of part (1) take place. Suppose that (a, b) = (1, 1) below, differentiating gc gives

gc (x ) =

ab c F (a + 1, b + 1; a + b + 1; x ) − F (2, 2; 3; x ) a+b 2



=

1 2ab F (a + 1, b + 1; a + b + 1; x ) F (2, 2; 3; x ) −c 2 a+b F (2, 2; 3; x )

=

1 F (2, 2; 3; x )[ξ (x ) − c]. 2



(3.10)

where ξ (x) is defined as in Lemma 3.4. When (a, b) ∈ E1 ∪ E2 , it follows from (3.10) and Lemma 3.4(2) that gc (x ) ≥ 0 if and only if c ≤ 1/B(a, b), gc (x ) ≤ 0 if and only if c ≥ 2ab/(a + b), and if 1/B(a, b) < c < 2ab/(a + b), then there exists x∗0 ∈ (0, 1 ) such that gc (x ) > 0 for x ∈ (0, x∗0 ), and gc (x ) < 0 for x ∈ (x∗0 , 1 ). Consequently part (2) follows. When (a, b) ∈ E3 , (3.10) together with the monotonicity of ξ (x) on (0, 1) in Lemma 3.4 shows that gc (x ) ≥ 0 if and only if c ≤ 2ab/(a + b), gc (x ) ≤ 0 if and only if c ≥ 1/B(a, b), and if 2ab/(a + b) < c < 1/B(a, b), then there exists x1 ∈ (0, 1) such that gc (x ) < 0 for x ∈ (0, x1 ), and gc (x ) > 0 for x ∈ (x1 , 1). Thus part (3) follows. In the remaining region (a, b) ∈ E4 , Lemma 3.4 shows that ξ (x) is piecewise monotone, namely, ξ (x) is decreasing and then increasing. Thus supx∈(0,1 ) ξ (x ) = max{2ab/(a + b), 1/B(a, b)}. Furthermore, if (a, b) ∈ E41 , then supx∈(0,1 ) ξ (x ) = 2ab/(a + b), so that from (3.10) we know that gc (x) is strictly decreasing on (0, 1) if c ≥ 2ab/(a + b), and if c = 1/B(a, b), then there exists x2 ∈ (0, 1) such that gc is strictly increasing in (0, x2 ), and strictly decreasing in (x2 , 1). Analogously, if (a, b) ∈ E42 , then supx∈(0,1 ) ξ (x ) = 1/B(a, b), so that gc (x) is strictly decreasing on (0, 1) if c ≥ 1/B(a, b), and if c = 2ab/(a + b), then there exists x3 ∈ (0, 1) such that gc is strictly decreasing in (0, x3 ), and strictly increasing in (x3 , 1).  Proof of Theorem 2.3. Since h(r) ≡ 1 when (a, b) = (1, 1 ), we need only consider the case that (a, b) = (1, 1) below. Let h1 (r ) = B(a, b)F (a, b; a + b, r ) + log(1 − r ) − R(a, b) and h2 (r ) = [(1 − r )/r] log[1/(1 − r )] = F (1, 1; 2; r ) + log(1 − r ), then simple computations lead to h(r ) = h1 (r )/h2 (r ),

h1 (1− ) = h2 (1− ) = 0, h1 (r ) = B(a, b) h2 (r ) = h1 (r ) = h2 (r )

(3.11)

1 ab F (a + 1, b + 1; a + b + 1; r ) − , a+b 1−r

1 1 F (2, 2; 3; r ) − , 2 1−r ab B a+b

(a, b)F (a, b; a + b + 1; r ) − 1 h3 (r ) ≡ , 1 h4 (r ) F (1, 1; 3; r ) − 1 2

h3 (1− ) = h4 (1− ) = 0, h3 (r ) = B(a, b) h4 (r ) =

(3.13)

(ab)2 F (a + 1, b + 1; a + b + 2; r ), (a + b)(a + b + 1 )

1 F (2, 2; 4; r ), 6

h3 (r ) 6a2 b2 = B(a, b)h5 (r ),  h4 (r ) (a + b)(a + b + 1 ) where

h5 (r ) = A†n =

(3.12)

∞ † n An r F (a + 1, b + 1; a + b + 2; r ) A† ( r ) ≡ † = n=0 , † n ∞ F (2, 2; 4; r ) B (r ) n=0 Bn r

( a + 1 )n ( b + 1 )n , (a + b + 2 )n n!

B†n =

( 2 )n ( 2 )n . (4 )n n!

(3.14)

(3.15)

(3.16)

56

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60 †

†

It is easy to check that the monotonicity of An /Bn depend on

Hn† = (ab + a + b − 3 )n + 4(ab − 1 )

(3.17)

and, by Lemma 3.3, the limiting value HA† ,B† (r ) at r = 1 is

lim HA† ,B† (r ) =

r→1−

R(2, 2 ) − R(a + 1, b + 1 ) 2 + R(a + 1, b + 1 ) =− . B(a + 1, b + 1 ) B(a + 1, b + 1 )

(3.18)

From (3.15)–(3.18) and Lemma 3.1 together with the similar argument in the proof of Theorem 2.1 we can conclude that h5 is strictly decreasing in (0, 1) if and only if (a, b) ∈ 1 ∪ 2 , and strictly increasing in (0, 1) if and only if (a, b) ∈ 3 . Besides, if (a, b) ∈ 4 , then there exists r0 ∈ (0, 1) such that h5 is strictly decreasing on (0, r0 ), and strictly increasing on (r0 , 1). Thus by (3.14), h3 /h4 has the same monotonicity as h5 ’s. It follows from (3.13) and the monotone properties of h3 /h4 on (0, 1) together with Lemma 3.2(1) that h3 /h4 is strictly decreasing on (0, 1) for (a, b) ∈ 1 ∪ 2 , and strictly increasing on (0, 1) for (a, b) ∈ 3 . Furthermore, if (a, b) ∈ 4 , then h3 /h4 is piecewise monotone on (0, 1). Note that h4 (r ) = F (2, 2; 4; r )/6 > 0 for all r ∈ (0, 1), and



lim+ Hh3 ,h4 (r ) = lim+

r→0

r→0



= lim+ r→0



−

h3 (r ) h4 (r ) − h3 (r ) h4 (r )





6a2 b2 F (a + 1, b + 1; a + b + 2; r ) 1 F (1, 1; 3; r ) − 1 B(a, b) (a + b)(a + b + 1 ) F (2, 2; 4; r ) 2

ab B(a, b)F (a, b; a + b + 1; r ) − 1 a+b





3a2 b2 ab B(a, b) + 1 B(a, b) − (a + b)(a + b + 1 ) a+b (a + b)(a + b + 1 ) − abB(a, b)(3ab + a + b + 1 ) = . (a + b)(a + b + 1 )

=−

Thus making use of Lemma 3.2(2), we clearly see that h3 /h4 is strictly increasing on (0, 1) if (a, b) ∈ 41 , and if (a, b) ∈ 42 , then there exists r1 such that h3 /h4 is strictly decreasing on (0, r1 ), and strictly increasing on (r1 , 1). Also, by (3.12), the monotone and piecewise monotone properties of h1 /h2 on (0, 1) follows. Analogously, employing Lemma 3.2(1) and (3.11), we can conclude that h1 /h2 , namely h, is strictly decreasing on (0, 1) for (a, b) ∈ 1 ∪ 2 , and strictly increasing on (0, 1) for (a, b) ∈ 3 ∪ 41 . Furthermore, if (a, b) ∈ 42 , then h1 /h2 is piecewise monotone on (0, 1). And by simple computation, one has

h2 (r ) =









1 1 1 1 F (1, 1; 3; r ) − 1 < F (1, 1; 3; 1 ) − 1 = 0 1−r 2 1−r 2

for all r ∈ (0, 1), and



lim Hh1 ,h2 (r ) = lim+

r→0+

r→0

= lim+ r→0



h1 (r ) h2 (r ) − h1 (r ) h2 (r ) ab B a+b



(a, b)F (a, b; a + b + 1; r ) − 1 [F (1, 1; 2; r ) + log(1 − r )] 1 F (1, 1; 3; r ) − 1  2

− [B(a, b)F (a, b; a + b; r ) + log(1 − r ) − R(a, b)]



abB(a, b) = 2 1− a+b =

 − B(a, b) + R(a, b)

[R(a, b) + 2 − B(a, b)](a + b) − 2abB(a, b) . (a + b )

Thus it follows from Lemma 3.2(2) that h is strictly increasing on (0, 1) if (a, b) ∈ ∗42 , and if (a, b) ∈ ∗∗ 42 , then there exists λ ∈ (0, 1) such that h is strictly decreasing on (0, λ), and strictly increasing on (λ, 1). So far all the monotone and piecewise monotone properties in Theorem 2.3 have been proved. Moreover, h(0+ ) = B(a, b) − R(a, b), and by l’Hôpital’ rule and (1.1), the limiting values h(r) at r = 1 is

h (r ) 6a2 b2 lim− h(r ) = lim− 3 = B(a, b) lim− h5 (r ) r→1 r→1 h (r ) r→1 (a + b)(a + b + 1 ) 4 =

6a2 b2 1/B(a + 1, b + 1 ) B(a, b) = ab. (a + b)(a + b + 1 ) 1/B(2, 2 )

So that inequalities (2.1)–(2.3) hold true. This completes the proof. 

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

Proof of Theorem 2.4. Clearly,



a1 , a2 , . . . , a p+1 b1 , b2 , . . . , b p

p+1 Fp

F (x ) =

   x

∞

F (1, 1; 2; x )

(a1 )n (a2 )n ...(a p+1 )n xn n! (b1 )n ...(b p )n ∞ ( 1 ) n ( 1 ) n x n n=0 (2 )n n!

n=0

=

57

,

If we denote

An =

(a1 )n (a2 )n . . . (a p+1 )n 1 ( b1 )n . . . ( b p )n n!

Bn =

( 1 )n ( 1 )n 1 , (2 )n n!

and

then

(a1 )n (a2 )n . . . (a p+1 )n (n + 1 ) An = ≡ Cn , Bn ( b1 )n . . . ( b p )n ( 1 )n (n + a1 )(n + a2 ) . . . (n + a p+1 )(n + 2 ) − (n + b1 )(n + b2 ) . . . (n + b p )(n + 1 )2 Cn+1 −1= . Cn (n + b1 )(n + b2 ) . . . (n + b p )(n + 1 )2 Let a p+2 = 2, b p+1 = 1 and b p+2 = 1, then expanding the numerator of the right hand of the above identity yields

(n + a1 )(n + a2 ) . . . (n + a p+1 )(n + 2 ) − (n + b1 )(n + b2 ) . . . (n + b p )(n + 1 )2 = n p+2 +

p+2  i1 =1



p+2 

=

bi1 n p+1 +

i1 =1







b i1 b i2 n p + · · · +

1≤i1


ai1 ai2 . . . ai p+1 n + a1 a2 . . . a p+1 a p+2

1≤i1
bi1 bi2 . . . bi p+1 n + b1 b2 . . . b p+1 b p+2

1≤i1


ai1 ai2 . . . aik −

1≤i1
k=2



ai1 ai2 n p + · · · +

1≤i1
− n p+2 + p+2 



ai1 n p+1 +

bi1 bi2 . . . bik n p−k+2 .

1≤i1
It is easy to show that

 1≤i1
1≤i1
 1≤i1
bi1 bi2 . . . bik−2

1≤i1
ai1 ai2 . . . aik −

1≤i1
+2



bi1 bi2 . . . bik−1 −



bi1 bi2 . . . bik−1

1≤i1
1≤i1




b i1 b i2 . . . b ik −



=

ai1 ai2 . . . aik−1

1≤i1
1≤i1
−



ai1 ai2 . . . aik + 2



bi1 bi2 . . . bik−1

1≤i1
1≤i1
−



b i1 b i2 . . . b ik −



ai1 ai2 . . . aik−1

1≤i1


=



ai1 ai2 . . . aik + 2

1≤i1
−

b i1 b i2 . . . b ik

1≤i1


=



ai1 ai2 . . . aik −



b i1 b i2 . . . b ik

1≤i1
ai1 ai2 . . . aik−1 −

 1≤i1
bi1 bi2 . . . bik−1

−

 1≤i1
Therefore, unitizing Lemma 3.1(1), the monotonicity of F on (0, 1) follows. Furthermore,

lim F (x ) = 1,

x→0+

and, from (1.1) and (1.8) one has

bi1 bi2 . . . bik−2 .

58

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60

1 [(R(a1 , a2 ) + M )(1 + O(1 − x )) − log(1 − x )(1 + O(1 − x ))] B − log(1 − x ) 1 = . B

lim F (x ) =

x→1−



Proof of Theorem 2.5. Define the function Gc on (0, 1) by



Gc (x ) = p+1 Fp

a1 , a2 , . . . , a p+1 b1 , b2 , . . . , b p

  c  x + x log(1 − x ), c ∈ R.

(3.19)

Then G(x ) = BG1/B (x ). Differentiating Gc yields



Gc (x ) =

⎡

=

⎢ 2a1 a2 . . . a p+1 1 F (2, 2; 3; x )⎢ ⎣ b1 b2 . . . b p 2

If we let



η (x ) =

  c  x − 2 F (2, 2; 3; x )    a1 + 1, a2 + 1, . . . , a p+1 + 1  x p+1 Fp b + 1, b + 1, . . . , b + 1

a1 a2 . . . a p+1 a1 + 1, a2 + 1, . . . , a p+1 + 1 p+1 Fp b1 + 1, b2 + 1, . . . , b p + 1 b1 b2 . . . b p

2a1 a2 . . . a p+1 b1 b2 . . . b p

p+1 Fp

1

p

2

F (2, 2; 3; x )

a1 + 1, a2 + 1, . . . , a p+1 + 1 b1 + 1, b2 + 1, . . . , b p + 1

   x

⎤ ⎥ ⎦

− c⎥.

x ∈ (0, 1 ).

,

F (2, 2; 3; x )

(3.20)

Then

lim

x→0+

η (x ) =

2a1 a2 . . . a p+1 , b1 b2 . . . b p

(3.21)

and by [ [12], Theorem 4], (b +1 )(b +1 )...(b +1 )

lim− η (x ) =

x→1

=

p 1 1 2 2a1 a2 . . . a p+1 (a1 +1)(a2 +1)...(a p+1 ) 1−z + O(1 ) (3 ) 1 b1 b2 . . . b p + O (1 )

(3.22)

1 . B

(3.22)

(2 )(2 ) 1−z

Note that

2a a . . . a p+1 η (x ) = 1 2 b1 b2 . . . b p

∞

n=0

(a1 +1 )n (a2 +1 )n ...(a p+1 +1 )n xn (b1 +1 )n (b2 +1 )n ...(b p +1 )n n! ∞ ( 2 ) n ( 2 ) n x n n=0 (3 )n n!

.

Denote

An =

(a1 + 1 )n (a2 + 1 )n . . . (a p+1 + 1 )n , (b1 + 1 )n (b2 + 1 )n . . . (b p + 1 )n n!

Bn =

2 ( 2 )n ( 2 )n ( 2 )n = , (3 )n n! (n + 2 )n!

then simple computation shows that the monotonicity of An /Bn depend on the sign of the polynomial (n + a1 + 1 )(n + a2 + 1 ) . . . (n + a p+1 + 1 )(n + 3 ) − (n + 1 + b1 )(n + 1 + b2 ) . . . (n + 1 + b p )(n + 2 )2 . Also we let a p+2 = 2, b p+1 = 1 and b p+2 = 1, then simplifying the polynomial gives

(n + a1 + 1 )(n + a2 + 1 ) . . . (n + a p+1 + 1 )(n + a p+2 + 1 ) − (n + 1 + b1 )(n + 1 + b2 ) . . . (n + 1 + b p )(n + b p+1 )(n + b p+2 ) =

p+2 



k=2

1≤i1


(ai1 + 1 )(ai2 + 1 ) . . . (aik + 1 ) −

(bi1 + 1 )(bi2 + 1 ) . . . (bik + 1 ) n p−k+2 .

1≤i1
With the similar argument in the proof of Theorem 2.4, the coefficient of the n p+2−k can be rewritten as



(ai1 + 1 )(ai2 + 1 ) . . . (aik + 1 ) −

1≤i1
=



1≤i1


(bi1 + 1 )(bi2 + 1 ) . . . (bik + 1 )

1≤i1
(ai1 + 1 )(ai2 + 1 ) . . . (aik + 1 ) + 3



1≤i1
(ai1 + 1 ) . . . (aik−1 + 1 )

M.-K. Wang et al. / Applied Mathematics and Computation 276 (2016) 44–60



−

1≤i1


=



(bi1 + 1 ) . . . (bik−1 + 1 ) − 4 (ai1 + 1 )(ai2 + 1 ) . . . (aik + 1 ) −



(bi1 + 1 )(bi2 + 1 ) . . . (bik + 1 )

1≤i1


(ai1 + 1 ) . . . (aik−1 + 1 ) − 4

1≤i1


(bi1 + 1 ) . . . (bik−2 + 1 )

1≤i1
1≤i1
−4

(bi1 + 1 ) . . . (bik−1 + 1 )

1≤i1
1≤i1
+3



(bi1 + 1 )(bi2 + 1 ) . . . (bik + 1 ) − 2





(ai1 + 1 ) . . . (aik−1 + 1 )

1≤i1
1≤i1
=



(ai1 + 1 )(ai2 + 1 ) . . . (aik + 1 ) + 3



−2

(bi1 + 1 ) . . . (bik−1 + 1 )

1≤i1
1≤i1
−



(bi1 + 1 )(bi2 + 1 ) . . . (bik + 1 ) − 2

59



(bi1 + 1 ) . . . (bik−1 + 1 )

1≤i1
(bi1 + 1 ) . . . (bik−2 + 1 ).

1≤i1
Thus it follows from (3.21) and (3.22) together with Lemma 3.1(1) that η is decreasing from (0, 1) onto (1/B, 2a1 a2 . . . a p+1 /[b1 b2 . . . b p ] ) if (a1 , . . . , a p+1 , b1 , . . . , b p ) ∈ 3 , and η is increasing in (0, 1) if (a1 , . . . , a p+1 , b1 , . . . , b p ) ∈ 4 , with range (2a1 a2 . . . a p+1 /[b1 b2 . . . b p ], 1/B ). Finally, when c = 1/B, (3.20) and the monotonicity and range of η on (0, 1) imply that G1/B (x ) is increasing in (0, 1) for (a1 , . . . , a p+1 , b1 , . . . , b p ) ∈ 3 , and Gc (x) is decreasing in (0, 1) for (a1 , . . . , a p+1 , b1 , . . . , b p ) ∈ 4 . Moreover,

lim Gc (x ) = 1 − 1/B,

x→0+

and by (1.8) we have

lim Gc (x ) =

x→1−

1 [R(a1 , a2 ) + M]. B

This completes the proof of Theorem 2.5.  Acknowledgments This research was supported by the Natural Science Foundation of China (Grant Nos. 61374086, 11371125), the Natural Science Foundation of Zhejiang Province (Grant no. LY13A010004) and the Natural Science Foundation of Hunan Province (Grant no. 14JJ2127) . Appendix. Figures See figures 1–4. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. [2] G.D. Anderson, R.W. Barnard, K.C. Richards, M.K. Vamanamurthy, M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 (5) (1995) 1713–1723. [3] G.D. Anderson, S.-L. Qiu, M.K. Vamanamurthy, M. Vuorinen, Generalized elliptic integrals and modular equations, Pacific J. Math. 192 (1) (2000) 1–37. [4] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen, Hypergeometric functions and elliptic integrals, in: H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publ. Co., Singapore-London, 1992. [5] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997. [6] G.E. Andrews, R. Askey, R. Roy, Special functions, Cambridge University Press, Cambridge, 1999. [7] R. Balasubramanian, S. Ponnusamy, M. Vuorinen, On hypergeometric functions and function spaces, J. Com. Appl. Math. 139 (2) (2002) 299–322. [8] R.W. Barnard, K. Pearce, K.C. Richards, A monotonicity property involving 3 F2 and comparisons of the classical approximations of elliptical arc length, SIAM J. Math. Anal. 32 (2) (2000) 403–419. [9] B.C. Berndt, Chapter 11 of Ramanujan’s second notebook, Bull. London. Math. Soc. 15 (4) (1983) 273–320. [10] B.C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989. [11] J.M. Borwein, P.B. Borwein, Pi and AGM, John Wiley & Sons, New York, 1987. [12] W. Buhring, Generalized hypergeometric functions at unit argument, Proc. Amer. Math. Soc. 114 (1) (1992) 145–153. [13] P. Deligne, G.D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, IHES. Publ. Math. 63 (1986) 5–89. [14] R.J. Evans, Ramanujan’s Second Notebook: Asymptotic Expansions for Hypergeometric Series and Related Functions, Academic Press, Boston, MA, 1988, pp. 537–560. Ramanujan Revisited (Urbana-Champaign, Ill., 1987). [15] R.J. Evans, D. Stanton, Asymptotic formulas for zero-balanced hypergeometric series, SIAM. J. Math. Anal. 15 (5) (1984) 1010–1020. [16] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010. [17] I. Pinelis, On l’hospital-type rules for monotonicity, JIPAM. J. Inequal. Pure Appl. Math. 7 (2) (2006) 1–19. Article 40 [18] S. Ponnusamy, M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (2) (1997) 278–301. [19] S. Ponnusamy, M. Vuorinen, Univalence and convexity properties of Gaussian hypergeometric functions, Rocky Mt. J. Math. 31 (1) (2001) 327–353. [20] S.-L. Qiu, M. Vuorinen, Landen inequalities for hypergeometric functions, Nagoya Math. J. 154 (1999) 31–56.

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