A result regarding monotonicity of the Gamma function

Monotonicity and Convexity for the Gamma Function

Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function θa,b(x) = [Γ(x + b)/Γ(x + a)]1/(b−a) − x is strictly convex and decreasing on (−a,∞) with θa,b(∞) = a+b−1 2 and θa,b(−a) = a, where Γ denotes the Euler’s gamma function.

Monotonicity and inequalities for the gamma function

Monotonicity Results for the Gamma Function

The function [Γ(x+1)] 1/x x+1 is strictly decreasing on [1,∞), the function [Γ(x+1)]1/x √ x is strictly increasing on [2,∞), and the function [Γ(x+1)] 1/x √ x+1 is strictly increasing on [1,∞), respectively. From these, some inequalities, for example, the Minc-Sathre inequality, are deduced, and two open problems posed by the second author are solved partially.

Insertion of a gamma-continuous function

A necessary and sufficient condition in terms of lower cut sets are given for the insertion of a $gamma-$continuous function between two comparable real-valued functions.

Complete Monotonicity of a Function Involving the Ratio of Gamma Functions and Applications

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result of Guo and Qi [Taiwanese J. Math. 7 (2003), no. 2, 239–247] and others. As applications, several inequalities involving the volume of the unit ball in R are derived, which refine, generalize a...