Monotonicity and sharp inequalities related to complete (p,q)-elliptic integrals of the first kind

Abstract

Monotonicity and sharp inequalities related to complete (p, q)-elliptic integrals of the first kind

On the maximum value of a confluent hypergeometric function

“…Under condition a > 0, several lower and upper bound inequalities for 2 F 1 (σ, a; b; x) have been derived in the literature using different approaches (e.g. [2][3][4]6,13,22] and references therein). For instance, in [13, Theorem 13], Luke gave the following two-sided bounds…”

“…On the other hand, in [22] the authors derived some inequalities for the Gauss hypergeometric function 2 F 1 (σ, a; b, x) when −1 < a < 0, 1 < b < 2, 0 < σ < 1, and x ∈ (0, 1). We remark that when a is a negative integer or zero, the estimate of the polynomial 2 F 1 (a, σ; b; x) has been considered in several papers from different point of views (see for instance [7,8] and references therein)…”