Inequalities and Monotonicity for the Ratio of Gamma Functions

On completely monotone of an arbitrary real parameter function involving the gamma function

“…Guo and Qi in [6] gave ðx þ y þ 1Þ=ðx þ y þ 2Þ as a lower bound for the left-hand side of the above inequality when y in its power satisfies y ¼ 0, here we give a best bound for the both sides of the general case of it.…”

An extension of an inequality for ratios of gamma functions

“…In [9,Theorem 2], the function [Γ (x + y + 1)/Γ (y + 1)] 1/x x + y + 1 (2) was proved to be decreasing with respect to x ≥ 1 for fixed y ≥ 0. Consequently, the inequality…”

A mononotonicity result of a function involving the gamma function

“…This short note is stimulated by an open problem posed in [1,5]. The origin, background, meanings, extensions, variants and generalizations of this problem can be found in [1,3,5,6] and some references therein, especially in [4].…”

“…Using Stirling's formula, for all nonnegative integers k and natural numbers n and m, an upper bound of the quotient of two geometrical means of natural numbers is established in [3] as follows: the lower bound of the quotient is given in [1,5,6] as follows (2) n + k + 1…”

“…As a generalization of inequality (1), an open problem was posed in [1,5]: Is it valid that for positive real numbers x, y and δ, we have (6) (F (x + y + 1)/Γ(y + 1)) 1/x (Γ(x + y + δ + 1)/Γ(y + 1)) 1/(x+δ) ≤…”