Certain Inequalities Involving the $q$-Deformed Gamma Function

Kwara Nantomah; Edward Prempeh

Certain Inequalities Involving the $q$-Deformed Gamma Function

Classical Analysis and ODEs 2015-07-01 v1

Authors: Kwara Nantomah , Edward Prempeh

Abstract

This paper is inspired by the work of J. S\'{a}ndor in 2006. In the paper, the authors establish some double inequalities involving the ratio $\frac{\Gamma_{q}(x+1)}{ \Gamma_{q} \left( x+\frac{1}{2}\right)}$ , where $\Gamma_{q}(x)$ is the $q$ -deformation of the classical Gamma function denoted by $\Gamma(x)$ . The method employed in presenting the results makes use of Jackson's $q$ -integral representation of the $q$ -deformed Gamma function. In addition, H\"{o}lder's inequality for the $q$ -integral, as well as some basic analytical techniques involving the $q$ -analogue of the psi function are used. As a consequence, $q$ -analogues of the classical Wendel's asymptotic relation are obtained. At the end, sharpness of the inequalities established in this paper is investigated.

Keywords

q-series and q-analogues inequalities gamma-convergence and homogenization

Cite

@article{arxiv.1506.09159,
  title  = {Certain Inequalities Involving the $q$-Deformed Gamma Function},
  author = {Kwara Nantomah and Edward Prempeh},
  journal= {arXiv preprint arXiv:1506.09159},
  year   = {2015}
}

Comments

Cite this article as: "K. Nantomah and E. Prempeh, Certain inequalities involving the q-deformed Gamma function, Problemy Analiza - Issues of Analysis, 3(22)(2015), No.1, In press."

Certain Inequalities Involving the $q$-Deformed Gamma Function

Abstract

Keywords

Cite

Comments

Related papers