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

Motivated by the work of Alzer and Richards \cite{ar}, here authors study the monotonicity and convexity properties of the function $$\Delta_{p,q} (r) = \frac{{E_{p,q}(r) - \left( {r'} \right)^p K_{p,q}(r) }}{{r^p }} - \frac{{E'_{p,q}(r) - r^p K'_{p,q}(r) }}{{\left( {r'} \right)^p }},$$ where $K_{p,q}$ and $E_{p,q}$ denote the complete $(p,q)$- elliptic integrals of the first and the second kind, respectively.


introduction
For 0 < r < 1 and r ′ = √ 1 − r 2 , the Legendre's complete elliptic integrals of the first and the second kind are defined by 1 − r 2 t 2 1 − t 2 dt, K(0) = π 2 = E(0), K(1) = ∞, E(1) = 0, K ′ = K ′ (r) = K(r ′ ) and E ′ = E ′ (r) = E(r ′ ), respectively. These integrals have played very crucial role in many branches of mathematics, for example, they helps us to find the length of curves and to express the solution of differential equations. The elliptic integrals of the first and the second kind have been extensive interest of the research for several authors, and many results have been established about these integrals in the literature For the monotonicity, convexity properties, asymptotic approximations, functional inequalities of these integrals and their relations with elementary functions, we refer the reader to see e.g. [5,Chapter 3] and the references therein. The Gaussian hypergeometric function is defined by Here (a, 0) = 1 for a = 0, and (a, n) is the Pochhammer symbol (a, n) = a(a + 1) · · · (a+n−1), for n ∈ N. The Gaussian hypergeometric function can be represented in the integral form as follows, For |s| < 1/2 and 1 ≤ |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).
In order to define the generalized complete (p, q)-elliptic integrals of the first and the second kind, we need to define the generalized sine function.
In 1998, Anderson, Qiu and Vamanamurthy studied the monotonicity and convexity property of the function E ′ − r K ′ by giving the following theorem.
Recently, Alzer and Richards [3] studied the properties of the additive counterpart of the above result, and proved the following theorem.
1.8. Theorem. 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

1.10.
Theorem. For all r, s ∈ (0, 1) and p, q > 1 satisfying the conditions (1) and (2) given in the above theorem, then we have

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].
This completes the proof.

3.2.
Remark. When p = q = 2, then it is easy to observe that Theorem 1.10 coincides with Proposition 1.6.