Complete monotonicity of a function involving the p-psi function and alternative proofs

Abstract

In the paper, the authors prove that the function x α ln px x+p+1 − ψ p (x) is completely monotonic on (0, ∞) if and only if α ≤ 1, where p ∈ N and ψ p (x) is the p-analogue of the classical psi function ψ(x).

On a conjecture of a logarithmically completely monotonic function

“…Krasniqi and Qi [11] proved that the function f p (t) = t • log pt t+p+1 − ψ p (t) is strictly completely monotonic on (0, ∞). The functions g(t) = − log t and h(t) = −γ • log t are also strictly completely monotonic on (0, 1).…”

Some Inequalities for the Extension of k-Gamma Function

“…In recent years, some extensions of the well known Euler's classical gamma function have been considered by several authors. Also many properties and inequalities concerning these functions have been examined; see for example, [1], [2], [3], [4], [5], [6] and [7]. The Chaudhry-Zubair extension *Corresponding author: E-mail: iege@adu.edu.tr; of the gamma function is defined as [8]…”

Extension of complete monotonicity results involving the digamma function

“…where, ψ p (z) is the p-digamma function and ψ k (z) is the k-digamma functions defined as follows (see [9], [4], [19], [21], [24]).…”