A class of logarithmically completely monotonic functions related to the q-gamma function and applications

Abstract

In this paper, the logarithmically complete monotonicity property for a functions involving q-gamma function is investigated for q ∈ (0, 1). As applications of this results, some new inequalities for the q-gamma function are established. Furthermore, let the sequence rn be defined by n! = √ 2πn(n/e) n e rn .We establish new estimates for Stirling's formula remainder rn.

Some characterizations of multiple selfdecomposability with extensions and an application to the Gamma function

“…if, and only if t ≥ 1 2 and r > 0 or t < 1 2 and r < 0. This generalizes several known ones in the litterature, c.f Alzer and Berg [2], Bertoin and Yor [7], Li and Chen [29], Mehrez [33], Pestana, Shanbhag, and Sreehari [36], for instance. If G t , G α 1 1,t .…”

“…we immediately retrieve the results of Li and Chen [29, Theorem 9] and also of Alzer and Berg [2, Theorem 3.5] on the functions G α . Mehrez [33,Theorem 1] obtained the same result in the context of the q-analogue of the function G α .…”

Some families of generalized Mathieu-type power series, associated probability distributions and related inequalities involving complete monotonicity and log-convexity

“…Before we present our main results in this section, we recall some standard definitions and basic facts. A non-negative function f defined on (0, ∞) is called completely monotonic if it has derivatives of all orders and (−1) n f (n) (x) ≥ 0, n ≥ 1 and x > 0 [ [15], [3], [13]]. This inequality is known to be strict unless f is a constant.…”