Inequalities Involving Gamma and Psi Functions (original) (raw)

Inequalities and monotonicity properties for gamma and q-gamma functions

arXiv (Cornell University), 2013

Most of the errors in the original paper had to do with saying that certain functions related to the q-gamma function were not completely monotonic. We discovered these errors through reading the paper Some completely monotonic functions involving the q-gamma function, by Peng Gao, http://arxiv.org/abs/1011.3303\. We also take the opportunity to correct some errors in other places including the statement and proof of Theorem 3.4.

Inequalities and monotonicity properties for the gamma functions

2001

We study the variation of the zeros of the Hermite function H\(t) with respect to the positive real variable A. We show that, for each non-negative integer n, H\(t) has exactly n + 1 real zeros when n < A ^ n + 1, and that each zero increases from-oo to oo as A increases. We establish a formula for the derivative of a zero with respect to the parameter A; this derivative is a completely monotonic function of A. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as well as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of H\(t). Similar results on zeros of certain confluent hypergeometric functions are given too. These specialize to results on the first, second, etc., positive zeros of Hermite polynomials.

Higher Monotonicity Properties of q-gamma and q-psi Functions

2013

Some typical results on complete monotonicity of a function f involving the gamma and q-gamma functions assert that f is completely monotonic for a certain range of values of a parameter, and −f is completely monotonic for another range, leaving a gap-an interval of parameter values where neither f nor −f is completely monotonic. We study this question for some examples involving q-gamma functions. In all cases considered here, the width of the gap decreases to 0 as q decreases from 1 to 0.

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

Applied Mathematics and Computation, 2013

Gamma, Psi (or Digamma) and Polygamma functions Double and multiple Gamma functions Riemann and Hurwitz (or generalized) Zeta functions Bohr-Mollerup theorem Harmonic numbers and the Stirling numbers of the first kind Euler-Mascheroni and Glaisher-Kinkelin constants Determinants of the Laplacians and Weierstrass canonical product forms Series involving the Zeta functions a b s t 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 C 2 . Here, in this paper, we aim at presenting several two-sided inequalities for the multiple Gamma functions C n ðn ¼ 2; 3; 4; 5Þ. In our investigation of these two-sided inequalities for the multiple Gamma functions C n ðn ¼ 2; 3; 4; 5Þ, we employ and extend a method based upon Taylor's formula and express log C n ð1 þ xÞ as series involving the Zeta functions. We also give a more convenient explicit form of the multiple Gamma functions C n ðn 2 NÞ; N being the set of positive integers. The main two-sided inequalities for the multiple Gamma functions C n ðn ¼ 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 C n when n=6.

Inequalities for the Psi and k-Gamma functions

Global Journal of Mathematical Analysis, 2014

In this paper, the authors establish some inequalities involving the Psi and k-Gamma functions. The procedure utilizes some monotonicity properties of some functions associated with the Psi and k-Gamma functions.