Monotonicity, convexity, and inequalities for the generalized elliptic integrals.

We provide the monotonicity and convexity properties and sharp bounds for the generalized elliptic integrals [Formula: see text] and [Formula: see text] depending on a parameter [Formula: see text], which contains an earlier result in the particular case [Formula: see text].

Introduction

For real numbers a, b, and c with  , the Gaussian hypergeometric function is defined by for , where denotes the shifted factorial function ,  , and for . It is well known that the function has many important applications in geometric function theory, theory of mean values, and several other contexts, and many classes of elementary functions and special functions in mathematical physics are particular or limiting cases of this function [1-10].

In what follows, we suppose , , and . The generalized elliptic integrals of the first and second kinds are defined as In the particular case , the generalized elliptic integrals and reduce to the complete elliptic integrals and , respectively. Recently, the Gaussian hypergeometric function and generalized elliptic integrals have been the subject of intensive research [2, 3, 5, 8, 11–30].

Anderson, Qiu, and Vamanamurthy [31] considered the monotonicity and convexity of the function One of the main results of [31] is the following theorem.

Theorem 1.1

The function is increasing and convex from onto . In particular, for . Both inequalities given in (1.4) are sharp as , whereas the second inequality is also sharp as .

Alzer and Richards [32] studied the corresponding properties of the additive counterpart and obtained the following theorem.

Theorem 1.2

The function is strictly increasing and strictly convex from onto . Moreover, for all , we have with the best constants and .

It is natural to extend Theorems 1.1 and 1.2 to the generalized elliptic integrals and . In this paper, we show the monotonicity and convexity of the functions and Moreover, we obtain sharp inequalities for them. If , then our results return to Theorems 1.1 and 1.2, which are contained in [31] and [32].

Preliminaries and lemmas

In this section, we give several formulas and lemmas to establish our main results stated in Section 1. First, let us recall some known results for .

The following formulas for the hypergeometric function can be found in the literature [33-35]: the differential formula the asymptotic limit and the contiguous relation where is the Euler gamma function.

Lemma 2.1

([2], Lemma 5.2)

Let . Then the function is increasing and convex from onto .

The following formulas were presented in [2]:

Lemma 2.2

([2], Lemma 2.3)

Let be an interval, and let . If both f, g are convex and increasing (decreasing), then the product is convex.

The following lemma follows from Theorem 1.7 in [1].

Lemma 2.3

For all , the function is a strictly decreasing automorphism of if and only if .

Lemma 2.4

The function is increasing from onto .

Proof

Let By the series expansion for we have By the definition of the generalized elliptic integrals of the first and second kinds (1.2) we have Since , , we have , and hence is an increasing function on . From this formula it is easy to see that . By Lemma 2.3 we have that . □

Lemma 2.5

([6], Lemma 2.1)

For , let be continuous on and differentiable on . Let on . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.

Main results and proofs

In this section, we present and prove two main theorems.

Theorem 3.1

The function in (1.6) is increasing and convex from onto . In particular, for with the best constant , . These two inequalities are sharp as , whereas the second inequality is sharp as .

Let Then By Lemma 2.1, , are positive increasing functions on , and hence is also an increasing function on . Since is a convex function by Lemma 2.1, the desired convexity of will follow from Lemma 2.2 if we prove that is a convex function on .

According to (2.6), we have where Obviously, . By Lemma 2.1 we get . Moreover, where is defined by (2.8). Hence, by Lemma 2.4 and Lemma 2.5, is decreasing, so that is increasing, and is convex on . □

Theorem 3.2

The function in (1.7) is strictly increasing and strictly convex from onto . Moreover, for all , we have with the best constants and . These two inequalities are sharp as , whereas the second inequality is sharp as .

Let By the series expansion for we obtain Then Using the differentiation formula (2.2), we have By formula (2.1),we get Using the contiguous relation (2.4), we take , , , and and obtain Hence, it follows from (3.6), (3.7), and the last formula that By the series expansion for we have Hence Through direct calculation we have Then we get . Thus is strictly convex on . According to (3.3) and (2.3), we have Applying Lemma 2.3 and (2.6), we have Because of , is increasing on , and . Then the monotonicity of on is obtained. It follows from the convexity of that, for ,  □

Corollary 3.3

Let Then we have for all .

By direct calculation we obtain Considering the positivity of and on , we have This means that is strictly increasing with respect to q. So we have Then the monotonicity of with respect to p is obtained, which leads to  □

Remark 3.4

Taking in Theorems 3.1 and 3.2, we get Theorems 1.1 and 1.2.