*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(5), 152-164

DOI: 10.12691/tjant-2-5-1

### Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity

**Feng Qi**^{1, }

^{1}Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

**Cite this paper:**

Feng Qi. Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity. *Turkish Journal of Analysis and Number Theory*. 2014; 2(5):152-164. doi: 10.12691/tjant-2-5-1

### Abstract

In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses 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.

**Keywords:**bound ratio of two gamma functions completely monotonic function logarithmically completely monotonic function divided difference gamma function q-gamma function psi function polygamma function inequality

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