Some two-sided inequalities for multiple Gamma functions and related results

There is an abundant literature on inequalities for the (Euler’s) Gamma function C and its various related functions. Yet, only very recently, several authors began to study inequalities for the (Barnes’) double Gamma function C2. Here, in this paper, we aim at presenting several two-sided inequalities for the multiple Gamma functions Cn ðn 1⁄4 2; 3; 4; 5Þ. In our investigation of these two-sided inequalities for the multiple Gamma functions Cn ðn 1⁄4 2; 3; 4; 5Þ, we employ and extend a method based upon Taylor’s formula and express logCnð1þ xÞ as series involving the Zeta functions. We also give a more convenient explicit form of the multiple Gamma functions Cn ðn 2 NÞ;N being the set of positive integers. The main two-sided inequalities for the multiple Gamma functions Cn ðn 1⁄4 2; 3; 4; 5Þ (which we have presented in this paper) are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple Gamma functions Cn when n=6. 2013 Elsevier Inc. All rights reserved.