Turán Type Inequalities for The $q$-exponential functions

Abstract

In this paper our aim is to deduce some sharp Turán type inequalities for the remainder q−exponential functions. Our results are shown to be a generalization of results which were obtained by Alzer [1].

Inequalities for series in q-shifted factorials and q-gamma functions

“…By Lemma 1 we see that a → h(a; x) is multiplicatively convex on (0, 1) for each fixed 0 < x < R h , where R h is the radius of convergence in (17). Next, we have: Corollary 4.1 Suppose α, β, µ > 0.…”

“…Finally, we furnish seven examples of q-hypergeometric functions satisfying our general theorems. Some results dealing with the Turán type inequalities for q-hypergeometric functions have been recently obtained by Baricz, Raghavendar and Swaminathan in [2,3] and Mehrez and Sitnik in [17,18]. In particular, our Theorem 1 can be seen as a far-reaching generalization of [2,Theorem 3.2], their connection explored in Example 2.…”

“…Then the function x → −∆ h (α, β; x) is multiplicatively convex on (0, R h ), while the function y → −∆ h (α, β; 1/y) is completely monotonic (and therefore log-convex ) on (1/R h , ∞). Corollary 4.2 Suppose {h n } n≥0 is a doubly positive sequence and h(a; x) is defined in (17). Then for all µ, β > 0, α ∈ N and 0 ≤ x < R h the following estimates hold: Γ q (µ + α)Γ q (µ + β) Γ q (µ)Γ q (µ + α + β) < h(q µ+α ; x)h(q µ+β ; x) h(q µ ; x)h(q µ+α+β ; x) ≤ 1 and Γ q (µ + α)Γ q (µ + β) Γ q (µ)Γ q (µ + α + β) − 1 h(q µ ; x)h(q µ+α+β ; x) < ∆ h (α, β; x)…”

“…Put h n = (a; q) n (b; q) n for n ∈ N 0 . Then for h(c; x) defined in (17) we have h(c; x) = 2 φ 1 (a, b; c; q; x), |x| < 1,…”