# Monotonicity Results For The Gamma Function

@article{Chen2002MonotonicityRF, title={Monotonicity Results For The Gamma Function}, author={Chao Chen and Feng Qi (祁锋)}, journal={Journal of Inequalities in Pure \& Applied Mathematics}, year={2002}, volume={4} }

The function 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 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.

## 41 Citations

MONOTONICITY AND CONVEXITY OF THE FUNCTION

- Mathematics
- 2004

For α > 0 a real number, the function x √ Γ(x+1) x+α √ Γ(x+α+1) is increasing with x ∈ (x0,∞) and logarithmically concave with x ∈ [1,∞), where x0 ∈ (0, 1) is a constant. Moreover, some monotonicity…

Complete Monotonicities of Functions Involving the Gamma and Digamma Functions

- Mathematics
- 2004

In the article, the completely monotonic results of the functions [Γ(x+ 1)]1/x, [Γ(x+α+1)]1/(x+α) [Γ(x+1)]1/x , [Γ(x+1)]1/x (x+1)α and [Γ(x+1)]1/x xα in x ∈ (−1,∞) for α ∈ R are obtained. In the…

A monotonicity property of the -function.

- Mathematics
- 2002

It is shown that the function x 7→ 1 + 1 x ln Γ(x+ 1)− ln(x+ 1) is strictly completely monotone on (−1,∞) and tends to one as x→ −1, to zero as x→∞. This property is derived from a suitable integral…

Monotonicity Results and Inequalities for the Gamma and Incomplete Gamma Functions

- Mathematics
- 2002

In the article, using the monotonicity and inequalities of the generalized weighted mean values with two parameters, we prove that the functions [ Γ(s)/Γ(r) ]1/(s−r) , [ Γ(s, x)/Γ(r, x) ]1/(s−r) and…

A Complete Monotonicity of the Gamma Function

- Mathematics
- 2004

The function 1 x ln Γ(x+1)−lnx+1 is strictly completely monotonic on (0,∞). The classical gamma function is usually defined for Re z > 0 by

Monotonicity of sequences involving geometric means of positive sequences with monotonicity and logarithmical convexity

- Mathematics
- 2006

Let f be a positive function such that x [ f (x + 1)/f (x)− 1 ] is increasing on [1,∞) , then the sequence { n √∏n i=1 f (i) / f (n + 1) }∞ n=1 is decreasing. If f is a logarithmically concave and…

SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS RELATED TO THE GAMMA FUNCTION

- Mathematics
- 2010

In this article, the logarithmically complete monotonicity of some functions such as 1 for fi 2 R on (i1;1) or (0;1) are obtained, some known results are recovered, extended and generalized.…

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In the article, using the monotonicity and inequalities of the generalized weighted mean values with two parameters, we prove that the functions [ Γ(s)/Γ(r) ]1/(s−r) , [ Γ(s, x)/Γ(r, x) ]1/(s−r) and…

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In this article, using Stirling’s formula, the series-expansion of digamma functions and other techniques, some inequalities and monotonicity concerning the ratio of gamma functions are obtained,…