On the complete monotonicity of quotient of gamma functions (original) (raw)

Bounds for the ratio of two gamma functions--From Gautschi's and Kershaw's inequalities to completely monotonic functions

2009

In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and divided differences of polygamma functions, and related monotonicity results.

Some inequalities for the ratio of gamma functions

Journal of Inequalities and Applications, 2015

In this paper, we present some sharp inequalities for the ratio of gamma functions. The main tool is the multiple-correction method formulated in (Cao et al.

On gamma function inequalities

Mathematics of Computation, 1986

We show that certain functions involving quotients of gamma functions are completely monotonie. This leads to inequalities involving gamma functions. We also establish the infinite divisibility of several probability distributions whose Laplace transforms involve quotients of gamma functions.

INEQUALITIES AND MONOTONICITY FOR THE RATIO OF GAMMA FUNCTIONS

2003

Let x > 0, y ≥ 0 be real numbers. The function f (x) = [Γp(x+y+1)/Γp(y+1)] 1 x x+y+1 is strictly decreasing and strictly logarithmically convex on (0, ∞). Moreover lim x→0 f (x) = e ψp(y+1) y+1 and x+y+1 x+y+2 < Γp(x+y+1)/Γp(y+1) 1 x Γp(x+y+2)/Γp(y+1) 1 x+1

Monotonicity results for the gamma function

2003

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.