## Generalized Hölder’s and Minkowski’s inequalities for Jackson’s $$q$$-integral and some applications to the incomplete $$q$$-Gamma function.(English)Zbl 1470.26034

Summary: We establish some generalized Hölder’s and Minkowski’s inequalities for Jackson’s $$q$$-integral. As applications, we derive some inequalities involving the incomplete $$q$$-Gamma function.

### MSC:

 26D15 Inequalities for sums, series and integrals 26E70 Real analysis on time scales or measure chains 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
Full Text:

### References:

 [1] Beckenbach, E. F.; Bellman, R., Inequalities, (1961), Berlin, Germany: Springer, Berlin, Germany · Zbl 0186.09606 [2] Campos, L. M., Generalized calculus with applications to matter and forces, (2014), New York, NY, USA: CRC Press, Taylor and Francis Group, New York, NY, USA · Zbl 1297.00007 [3] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, (1952), London, UK: Cambridge University Press, London, UK · Zbl 0047.05302 [4] Jackson, F. H., On a q-definite integrals, Quarterly Journal of Pure and Applied Mathematics, 41, 193-203, (1910) · JFM 41.0317.04 [5] Krasniqi, V.; Mansour, T.; Shabani, A. S., Some inequalities for q-polygamma function and {\itζq}-Riemann zeta functions, Annales Mathematicae et Informaticae, 37, 95-100, (2010) · Zbl 1224.33010 [6] Tunc, M.; Gov, E., Some Integral Inequalities Via (p,q)-Calculus on Finite Intervals, RGMIA Research Report Collection, 19, (2016) [7] El-Shahed, M.; Salem, A., On q-analogue of the Incomplete Gamma Function, International Journal of Pure and Applied Mathematics, 44, 5, 773-780, (2008) · Zbl 1246.05020 [8] Salem, A., A q-analogue of the exponential integral, Afrika Matematika, 24, 2, 117-125, (2013) · Zbl 1353.41007 [9] Nantomah, K.; Nasiru, S., Inequalities for the m-th derivative of the (q, k)-Gamma function, Moroccan Journal of Pure and Applied Analysis, 3, 1, 63-69, (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.