$\bar{q}$-Inequalities on quantum integral

Necmettin Alp; Mehmet Zeki Sarıkaya

doi:10.26637/MJM0804/0121

Abstract

In this paper, we present $\bar{q}$-Young integral inequality, $\bar{q}$-Hölder integral inequality, $\bar{q}$-Minkowski integral inequality and $\bar{q}$-Ostrowski type integral inequalities for new definition of $q$-integral which is showed $\bar{q}$-integral.

Keywords:

Ostrowski inequality, Young, H¨ older and Minkowski integral inequalities, convex functions, $\bar{q}$-integrals

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Necmettin Alp Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-TURKEY. https://orcid.org/0000-0002-9123-6187
Mehmet Zeki Sarıkaya Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-TURKEY.

Pages: 2035-2044
Date Published: 01-10-2020
Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)

V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.

N. Alp, M. Z. Sarikaya, M. Kunt and İ. İşcan, $q$-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, Journal of King Saud University - Science, (2018) 30, 193-203.

N. Alp, M. Z. Sarikaya, A New Definition and Properties of Quantum Integral Which calls $bar{q}$-Integral, Konuralp Journal of Mathematics, Volume 5 No. 2 pp.146-159 (2017).

M. Kunt, I. Iscan, N. Alp and M. Z. Sarikaya, $(p, q)$ Hermite-Hadamard inequalities and $(p, q)$-estimates for midpoint type inequalities via convex and quasi-convex functions, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 112(4), 969-992 (2018).

H. Gauchman, Integral inequalities in $q$-calculus, Comput. Math. Appl., 2004, 47: 281-300.

C. BorellInverse, Hölder inequalities in one and several dimensions, J. Math. Anal. Appl., 41(2)(1973), 300-312.

D.S. Mitrinovic, J.E. Pecaric and A.M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht, 1994.

M.A. Noor, K.I. Noor and M.U. Awan, Some Quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput., 251, 675-679 (2015).

M.A. Noor, K.I. Noor and M.U. Awan, Quantum Ostrowski inequalities for $q$-differentiable convex functions, J. Math. Inequlities, 10.4 (2016): 1013-1018.

H. Ogunmez and U.M. Ozkan, Fractional quantum integral inequalities, J. Inequal. Appl., 2011., 2011: Article ID 787939.

G.H. Hardy, J.E. Littlewood and G. Polya, Minkowski's' Inequality and Minkowski's Inequality for Integrals, 2.11, 5.7, and 6.13 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 30-32, 123, and 146$150,1988$.

J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013, 2013:282.

W. H. Young, On classes of summable functions and their Fourier series, Proc. Royal Soc., Series (A), 87 (1912) 225-229.

F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 1910, 41, 193-203.

F.H. Jackson, q-form of Taylor's theorem, Messenger Math., 39 (1909) 62-64.

H. Exton, q-hypergeometric functions and applications, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1983.

Al-Salam W A, q-Bernoulli Numbers and Polynomials, Math. Nachr., 17 (1959), 239-260.

Al-Salam W A and Verma A, A Fractional Leibniz q-Formula, Pacific J. Math., 60, Nr. 2, (1975), 1-9.

M. S. Stankoyić, P. M. Rajkoyić, S. D. Marinkoyić, Inequalities which include $q$-integrals, Serbian Academy of Sciences and Arts, No. 31 (2006), pp. 137-146.