Geodesic
Certain inequalities involving the $q$-deformed Gamma function
Problemy analiza, Tome 4 (2015) no. 1, pp. 57-65.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper is inspired by the work of J. Sándor 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, Hölder'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: Gamma function, $q$-deformed Gamma function, $q$-integral, inequality. 
@article{PA_2015_4_1_a3,
     author = {K. Nantomah and E. Prempeh},
     title = {Certain inequalities involving the $q$-deformed {Gamma} function},
     journal = {Problemy analiza},
     pages = {57--65},
     publisher = {mathdoc},
     volume = {4},
     number = {1},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2015_4_1_a3/}
}
TY  - JOUR
AU  - K. Nantomah
AU  - E. Prempeh
TI  - Certain inequalities involving the $q$-deformed Gamma function
JO  - Problemy analiza
PY  - 2015
SP  - 57
EP  - 65
VL  - 4
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2015_4_1_a3/
LA  - en
ID  - PA_2015_4_1_a3
ER  - 
%0 Journal Article
%A K. Nantomah
%A E. Prempeh
%T Certain inequalities involving the $q$-deformed Gamma function
%J Problemy analiza
%D 2015
%P 57-65
%V 4
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2015_4_1_a3/
%G en
%F PA_2015_4_1_a3
K. Nantomah; E. Prempeh. Certain inequalities involving the $q$-deformed Gamma function. Problemy analiza, Tome 4 (2015) no. 1, pp. 57-65. http://geodesic.mathdoc.fr/item/PA_2015_4_1_a3/

[1] Jackson F. H., “On a $q$-definite integrals”, Quarterly Journal of Pure and Applied Mathematics, 41 (1910), 193–203 | Zbl

[2] Askey R., “The $q$-Gamma and $q$-Beta Functions”, Applicable Anal., 8:2 (1978), 125–141 | DOI | MR | Zbl

[3] Chung W. S., Kim T., Mansour T., “The $q$-deformed Gamma function and $q$-deformed Polygamma function”, Bull. Korean Math. Soc., 51:4 (2014), 1155–1161 | DOI | MR | Zbl

[4] Ege I., Yýldýrým E., “Some generalized equalities for the $q$-gamma function”, Filomat, 12:6 (2012), 1221–1226 | DOI | MR

[5] Elmonster H., Brahim K., Fitouhi A., “Relationship between characterizations of the $q$-Gamma function”, Journal of Inequalities and Special Functions, 3:4 (2012), 50–58 | MR

[6] Shabani A. S., “Generalization of some inequalities for the $q$-gamma function”, Annales Mathematicae et Informaticae, 35 (2008), 129–134 | MR | Zbl

[7] Lew J., Frauenthal J., Keyfitz N., On the average distances in a circular disc. Mathematical Modeling: Classroom Notes in Applied Mathematics, SIAM, Philadelphia, 1987

[8] Sándor J., “On certain inequalities for the Gamma function”, RGMIA Res. Rep. Coll., 9:1 (2006), 11 | MR

[9] Wendel J. G., “Note on the gamma function”, Amer. Math. Monthly, 55 (1948), 563–564 | DOI | MR

[10] Qi F., Luo Q. M., Banach Journal of Mathematical Analysis, 6:2 (2012), Bounds for the ratio of two Gamma functions — from Wendel's and related inequalities to logarithmically completely monotonic functions | MR

[11] Ismail M. E. H., Muldoon M. E., Inequalities and monotonicity properties for Gamma and $q$-Gamma functions, arXiv: 1301.1749v1 | MR

[12] Mortici C., “A sharp inequality involving the psi function”, Acta Universitatis Apulensis, 22 (2010), 41–45 | MR

[13] Qi F., “Bounds for the ratio of two Gamma functions”, Journal of Inequalities and Applications, 2010, 493058 | MR | Zbl