On a function involving generalized complete (p, q)-elliptic integrals

Motivated by the work of Alzer and Richards (Anal Math 41:133–139, 2015), here authors study the monotonicity and convexity properties of the function Δp,q(r)=Ep,q(r)-r′pKp,q(r)rp-Ep,q′(r)-rpKp,q′(r)r′p,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _{p,q} (r) = \frac{{E_{p,q}(r) - \left( {r'} \right) ^p K_{p,q}(r) }}{{r^p }} - \frac{{E'_{p,q}(r) - r^p K'_{p,q}(r) }}{{\left( {r'} \right) ^p }}, \end{aligned}$$\end{document}where Kp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{p,q}$$\end{document} and Ep,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{p,q}$$\end{document} denote the complete (p, q)-elliptic integrals of the first and second kind, respectively.

The Gaussian hypergeometric function can be represented in the integral form as follows: For |s| < 1/2 and 0 ≤ |r | < 1, the complete elliptic integrals of the first and the second kind were slightly generalized by Borwein and Borwein [12] as follows: Note that K 0 (r ) = K(r ) and E 0 (r ) = E(r ).
To define the generalized complete ( p, q)-elliptic integrals of the first and the second kind, we need to define the generalized sine function.
The eigenfunction sin p,q of the so-called one-dimensional ( p, q)-Laplacian problem [20] is known as the generalized sine function with two parameters p, q > 1 in the literature (see, [9,10,14,21,24,[28][29][30]), and defined as the inverse function of Also the generalized π is defined as which is the generalized version of the celebrated formula of π proved by Salamin [27] and Brent [13] in 1976.
Here B(., .) denotes the classical beta function. For all p, q ∈ (1, ∞), r ∈ (0, 1) and r = (1 − r p ) 1/ p , the generalized complete ( p, q)-elliptic integrals of the first and the second kind are defined by respectively. Applying the integral representation formula (1.2), the generalized complete ( p, q)-elliptic integrals can be expressed in terms of hypergeometric functions as follows: and see [11]. For p = q, we write K p, p = K p . Note that K 2 = K and E 2 = E. It is worth to mention that Takeuchi [24] proved that for |s| < 1/2 and p = 2/(2s + 1).
In 1998, Anderson, Qiu and Vamanamurthy [4, Theorem 1.14] studied the monotonicity and convexity property of the function by giving the following theorem.

Theorem 1.7
The function f (r ) is increasing and convex from (0, 1) onto (π/4, 4/π). In particular, for r ∈ (0, 1). These two inequalities are sharp as r → 0, while the second inequality is also sharp as r → 1. Recently, Alzer and Richards [3] studied the properties of the additive counterpart of the above result, and proved the following theorem.

Proposition 1.8
The function (r ) is strictly increasing and strictly convex from (0, 1) onto (π/4−1, 1−π/4). Moreover, for all r ∈ (0, 1), one has with best possible constants α = 0 and β = 2 − π 2 = 0.42920 . . . . It is natural to extend the result of Alzer and Richards in terms of generalized complete ( p, q)− elliptic integrals of the first and second kind. We generalize their function by and state the following theorem. Theorem 1.10 For p, q > 1, the function p,q is strictly increasing and strictly convex from (0, 1) onto , if the following conditions hold: Moreover, for all r ∈ (0, 1), we have with best possible constants α 1 = 0 and

Lemmas
In this section, we give few lemmas which will be used in the proof of the theorems. Moreover, we will use same method for proving our theorems as it is applied in [3].

Lemma 2.1 Write
Proof Using the formula (1.1), we obtain

Lemma 2.3
For p, q > 1 and r ∈ (0, 1), we have Proof By definition, it is easy to see that Again, using the following identity

Proof of the main result
Proof of Theorem 1.10 Using formulas (1.5), (1.6) and letting a = 1/q, b = 1/ p in (2.2), we have p,q (r ) = H 1/q,1/ p (r ) − H 1/q,1/ p (r ) Applying the following derivative formula and 1 η p,q r p−2 p,q (r ) By utilizing the following identity (see [3]) and letting a = 1 + 1 q , b = 2 − 1 p , z = 1 − r p , we get 1 η p,q r p−2 p,q (r ) Now by Lemma 2.6 we have 1 η p,q r p−2 p,q (r ) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.