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.

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.

Inequalities for the qqq-gamma and related functions

arXiv: Number Theory, 2018

We consider convexity and monotonicity properties for some functions related to the qqq-gamma function. As applications, we give a variety of inequalities for the qqq-gamma function, the qqq-digamma function psiq(x)\psi_q(x)psiq(x), and the qqq-series. Among other consequences, we improve a result of Azler~and~Grinshpan about the zeros of the function psiq(x)\psi_q(x)psiq(x). We use qqq-analogues for the Gauss multiplication formula to put in closed form members of some of our inequalities.