Some inequalities and monotonicity properties associated with the gamma and psi functions and the Barnes G-function

Abstract

In this paper, several properties associated with inequalities and the logarithmically complete monotonicity of functions related to the gamma and psi functions and the Barnes G-function are obtained. Relevant connections of the results presented here with those derived in earlier works are also pointed out.

Keywords:

• gamma function
• psi function
• polygamma functions
• Barnes's G-function
• completely monotonic functions
• logarithmically completely monotonic functions
• absolutely monotonic functions
• inequalities and monotonicity properties
• Riemann's zeta function
• Hurwitz's (or generalized) zeta function

2010 Mathematics Subject Classification :

• Primary: 33B15
• Secondary: 11M06
• 11M35
• 26A48
• 26A51

Acknowledgements

The first-named author was supported partially by the Natural Scientific Research Subsidy Planning Project of the Education Department of the Henan Province of the People's Republic of China under Grant 2008A110007.