Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

Journal of Inequalities and Applications

Ahmed Salem¹  Faris Alzahrani¹  Moustafa El-Shahed² 

[1] Department of Mathematics, Faculty of Science, King Abdulaziz University;Unaizah Faculty of Arts and Sciences, Qassim University;

关键词: q-Gamma function;    q-Digamma function;    Inequalities;    Completely monotonic function;   

DOI  :  10.1186/s13660-021-02642-7

来源: DOAJ

【 摘 要 】

Abstract In the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions Γ q ( x + 1 ) / Γ q ( x + s ) ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ for all real number s and 0 < q ≠ 1 $0< q\neq1$ in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all 0 < s < 1 $0< s<1$ when letting q → 1 $q\to 1$ and a new inequality for all s > 1 $s>1$ .

【 授权许可】

Unknown