On (p, q) and (q, k)-extensions of a double-inequality bounding a ratio of Gamma functions

Year 2016, Volume: 4 Issue: 2, 23 - 28, 30.10.2016

Kwara Nantomah , Edward Prempeh Stephen Boakye Twum

https://doi.org/10.36753/mathenot.421447

Abstract

In this paper, the authors present the (p, q) and (q, k)-extensions of a double inequality involving a ratio of
Gamma functions. The method is based on some monotonicity properties of certain functions associated
with the (p, q) and (q, k)-extensions of the Gamma function.

Keywords

Gamma function , psi function , inequality , (pˏ q)-extension , (qˏk)-extension

References

• [1] C. Alsina and M. S. Tomás, A geometrical proof of a new inequality for the gamma function, J. Ineq. Pure Appl. Math., 6(2)(2005), Art. 48.
• [2] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
• [3] R. Díaz and E. Pariguan, On hypergeometric functions and Pachhammer k-symbol, Divulgaciones Matemtícas, 15(2)(2007), 179-192.
• [4] R. Díaz and C. Teruel, q, k-generalized gamma and beta functions, J. Nonlin. Math. Phys., 12(2005), 118-134.
• [5] F. H. Jackson, On a q-Definite Integrals, Quarterly Journal of Pure and Applied Mathematics, 41(1910), 193-203.
• [6] V. Krasniqi and F. Merovci, Some Completely Monotonic Properties for the (p, q)-Gamma Function, Mathematica Balkanica, New Series, 26(2012), Fasc. 1-2.
• [7] K. Nantomah, Some Inequalities for the Ratios of Generalized Digamma Functions, Advances in Inequalities and Applications, 2014(2014), Article ID 28.
• [8] K. Nantomah and M. M. Iddrisu, The k-analogue of some inequalities for the Gamma function, Electron. J. Math. Anal. Appl., 2(2)(2014), 172-177.
• [9] K. Nantomah, E. Prempeh and S. B. Twum, The (q, k)-extension of some Gamma function inequalities, Konuralp Journal of Mathematics, 4(1)(2016), 148-154.
• [10] N. V. Vinh and N. P. N. Ngoc, An inequality for the Gamma Function, International Mathematical Forum, 4(28)(2009), 1379-1382.
• [11] N. P. N. Ngoc, N. V. Vinh and P. T. T. Hien, Generalization of Some Inequalities for the Gamma Function, Int. J. Open Problems Compt. Math., 2(4)(2009), 532-535.
• [12] J. Zhang and H. Shi, Two double inequalities for k-gamma and k-Riemann zeta functions, Journal of Inequalities and Applications, 2014, 2014:191.