MONOTONICITY RESULTS AND ASSOCIATED INEQUALITIES FOR K-GAMMA FUNCTION

Abstract

In this work, we find-analogue of Gautschi inequality and obtain some monotonicity results for-gamma function. By the aid of these, we also get some more inequalities.This study is inspired bythe work of Neumann in 2011. In the study, weestablish some double inequalities involving the ratio𝛤𝑘(𝑛+𝑠)𝛤𝑘(𝑛+𝑘), where 𝛤𝑘is the 𝑘-analogue of Euler’s gamma function. Some monotonicity results involvingk-gamma functionare found. By the aid of these results, some inequalitiessuchas[Γ𝑘(𝑘+𝜎)]𝑘𝜎≤∑[Γ𝑘(𝑘+𝑥𝑖)]𝑘𝑥𝑖𝑛𝑖=1for 𝑥𝑖>0, 1≤𝑖≤𝑛, where 𝜎=𝑥1+𝑥2+⋯+𝑥𝑛are valid.