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Sharp bounds for the q-gamma function in terms of the Lambert W function

  • Ahmed Salem
Article
  • 47 Downloads

Abstract

In this paper, the function \(F_a(x;q)\) is defined in terms of logarithmic of q-gamma function for all reals xa and q with \(q>0\). The values of a in which the function \(F_a(x;q)\) is completely monotonic function are determined and it turns out that these values depend on the Lambert W function. As a consequence, sharp upper and lower bounds for the q-gamma, q-digamma and q-polygamma functions for all positive real q in terms of the Lambert W function are provided. Analytically and numerically, our results are compared with previous results. Furthermore, best bounds for the Lambert W function are provided.

Keywords

q-Gamma function q-Polygamma functions Lambert W function Inequalities Completely monotonic function 

Mathematics Subject Classification

33D05 26D07 26A48 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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