Monotonicity Results for the Gamma Function

Volume 4, Issue 2, 2003

Article 44

MONOTONICITY RESULTS FOR THE GAMMA FUNCTION

DEPARTMENT OF APPLIED MATHEMATICS AND INFORMATICS,
JIAOZUO INSTITUTE OF TECHNOLOGY,
JIAOZUO CITY, HENAN 454000,
THE PEOPLE'S REPUBLIC OF CHINA.
E-Mail: qifeng@jzit.edu.cnURL: http://rgmia.vu.edu.au/qi.html

Received 01 June, 2002; Accepted 01 May, 2003.
Communicated by: A. Laforgia

ABSTRACT. The function $\frac{[\Gamma(x+1)]^{1/x}}{x+1}$ is strictly decreasing on $% [1,\infty)$ , the function $\frac{[\Gamma(x+1)]^{1/x}}{\sqrt{x}}$ is strictly increasing on $[2,\infty)$ , and the function $\frac{[\Gamma(x+1)]^{1/x}}{% \sqrt{x+1}}$ is strictly increasing on $[1,\infty)$ , 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.

Key words:
Gamma function, Monotonicity, Inequality.

2000 Mathematics Subject Classification:
Primary 33B15; Secondary 26D07.

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