MONOTONICITY RESULTS AND ASSOCIATED INEQUALITIES FOR K-GAMMA FUNCTION

Yıl 2019, Cilt: 20 Sayı: 1, 104 - 111, 01.01.2019

İnci Ege Emrah Yıldırım

https://doi.org/10.18038/aubtda.438249

Öz

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.

Anahtar Kelimeler

Gamma function, k-gamma function, monotonicity

Kaynakça

• Gautschi W. Some elementary inequalities relating to the gamma and incomplete gamma function. Journal of Mathematics and Physics 1959; 38-1: 77-81.
• Neuman E. Some inequalities for the gamma function. Appl. Math. Comput. 2011; 218-8: 4349-4352.
• Diaz R, Pariguan E. On hypergeometric functions and Pachhammer k-symbol. Divulg. Mat. 2007; 15-2: 179-192.
• Krasniqi V. Inequalities and monotonicity for the ration of k -gamma function. Scientia Magna. 2010; 6-1: 40-45.
• Gradshteyn, I. S., Ryzhik, I. M. Table of integrals, series, and products. Academic Press, 2014.