Document Type : Research Paper


1 Department of Mathematics, Shah Abdul Latif University Khairpur-66020, Pakistan.

2 Department of Mathematics, University of Karachi, University Road, Karachi-75270, Pakistan.

3 Department of Basic Sciences, Mathematics and Humanities, Dawood University of Engineering and Technology, M. A Jinnah Road, Karachi-74800, Pakistan.


In this paper, we are introducing for the first time a generalized class named the class of $(\alpha,\beta,\gamma,\delta)-$convex functions of mixed kind. This generalized class contains many subclasses including the class of $(\alpha,\beta)-$convex functions of the first and second kind, $(s,r)-$convex functions of mixed kind, $s-$convex functions of the first and second kind, $P-$convex functions, quasi-convex functions and the class of ordinary convex functions. In addition, we would like to state the generalization of the classical Ostrowski inequality via fractional integrals, which is obtained for functions whose first derivative in absolute values is $(\alpha,\beta,\gamma,\delta)-$ convex function of mixed kind. Moreover, we establish some Ostrowski-type inequalities via fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are $(\alpha,\beta,\gamma,\delta)-$ convex functions of mixed kind using different techniques including H\"older's inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, the applications of special means will also be discussed.


Main Subjects

[1] H. Ahmad, M. Tariq, S.K. Sahoo, S. Askar, A.E. Abouelregal and K.M. Khedher, Refinements of Ostrowski-type integral inequalities involving Atangana--Baleanu fractional integral operator, Symmetry., 13, (2021), article: 2059.
[2] M. Alomari, M. Darus, S.S. Dragomir and P. Cerone, Ostrowski-type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23, (2010), pp. 1071-1076.
[3] A. Arshad and A.R. Khan, Hermite$-$Hadamard$-$Fejer type integral inequality for $s-p-$convex of several kinds, Transylv. J. Math. Mech., 11 (2019), pp. 25-40.
[4] E.F. Beckenbach, Convex, Bull. Amer. Math. Soc., 54 (1948), pp. 439-460.
[5] B. Benaissa, and A. Senouci, New integral inequalities relating to a general integral operators through monotone functions, Sahand Commun. Math. Anal., 19(1), (2022), pp. 41-56.
[6] W.W. Breckner, Stetigkeitsaussagen Fur eine klasse verallgemeinerter konvexer funktionen in topologischen linearen raumen., Publ. Inst. Math. Univ. German., 23 (1978), pp. 13-20.
[7] E. Set, New inequalities of Ostrowski-type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), pp. 1147-1154.
[8] M.J.V. Cortez and J.E. Hernández, Ostrowski and Jensen-type inequalities via $ (s, m) $-convex in the second sense, Bol. Soc. Mat. Mex., 26 (2020), pp. 287-302.
[9] B. Daraby, A. Khodadadi, and A. Rahim, Godunova type inequality for Sugeno integral, Sahand Commun. Math. Anal., 19(4), (2022), pp. 39-50.
[10] B. Daraby, Generalizations of some inequalities for Sugino integrals, Sahand Commun. Math. Anal., 19(3), (2022), pp. 141-168.
[11] S.S. Dragomir, A companion of Ostrowski's inequality for functions of bounded variation and applications,     Int. J. Nonlinear Anal. Appl., 5 (2014), pp. 89-97.
[12] S.S. Dragomir, The functional generalization of Ostrowski inequality via montgomery identity, Acta Math. Univ. Comenianae., 1 (2015), pp. 63-78.
[13] S.S. Dragomir, On the Ostrowski's integral inequality for mappings with bounded variation and applications, Math. Inequal. Appl., 4 (2001), pp. 59-66.
[14] S.S. Dragomir, Integral inequalities of Jensen type for $\lambda$-convex, Res. Rep., Kitakyushu Coll. Technol., 17 (2014).
[15] S.S. Dragomir, Refinements of the generalised trapozoid and Ostrowski inequalities for functions of bounded variation, Arch. Math., 91 (2008), pp. 450-460.
[16] S.S. Dragomir and N. S. Barnett, An Ostrowski-type inequality for mappings whose second derivatives are bounded and applications, J. Indian Math. Soc., 66 (1999), pp. 237-245.
[17] S.S. Dragomir, P. Cerone, N.S. Barnett and J. Roumeliotis, An inequality of the Ostrowski-type for double integrals and applications for cubature formulae, Tamsui Oxf. J. Inf. Math. Sci., 16 (2000), pp. 1-16.
[18] S.S. Dragomir, P. Cerone and J. Roumeliotis, A new generalization of Ostrowski integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Appl. Math. Lett., 13 (2000), pp. 19-25.
[19] S.S. Dragomir, J. Pecaric and L. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), pp. 335-341.
[20] A. Ekinci, Klasik esitsizlikler yoluyla konveks fonksiyonlar icin integral esitsizlikler, Ph.D. Thesis: Atat\"urk University, 2014.
[21] E.K. Godunova and V.I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, Numerical Mathematics and Mathematical Physics, 1985.
[22] N. Irshad, A.R. Khan and A. Nazir, Extension of Ostrowki type inequality via moment generating gunction, Adv. Ineq. Appl, 2 (2020), pp. 1-15.
[23] N. Irshad, A.R. Khan and M.A. Shaikh, Generalization of weighted Ostrowski inequality with applications in numerical integration, Adv. Ineq. Appl., 7 (2019), pp. 1-14.
[24] N. Irshad, A.R. Khan and M.A. Shaikh, Generalized weighted Ostrowski-Gruss type inequality with applications, Glob. J. Pure Appl. Math., 15 (2019), pp. 675-692.
[25] N. Irshad and A.R. Khan, On weighted Ostrowski Gruss inequality with applications, Transylv. J. Math. Mech., 10 (2018), pp. 15-22. 
[26] N. Irshad and A.R. Khan, Generalization of Ostrowski inequality for differentiable functions and its applications to numerical quadrature rules, J. Math. Anal. Appl., 8 (2017), pp. 79-102.
[27] A. Kashuri, B. Meftah, P.O. Mohammed, A.A. Lupaş, B. Abdalla, Y.S. Hamed and T. Abdeljawad, Fractional weighted Ostrowski-type inequalities and their applications, Symmetry., 13, (2021), art: 968.
[28] M. Matłoka, On Ostrowski-type inequalities via fractional integrals with respect to another function, Int. J. Appl. Nonlinear Sci., 13 (2020), pp. 100-106 
[29] L. Nasiri, and M. Shams, The generalized inequalities via means and positive linear appings, Sahand Commun. Math. Anal., 19(2), (2022), pp. 133-148. doi: 10.22130/scma.2022.544128.1028.
[30] M.A. Noor and M.U. Awan, Some integral inequalities for two kinds of convexities via fractional integrals, Transylv. J. Math. Mech., 5 (2013), pp. 129-136.
[31] A. M. Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv., 10 (1938), pp. 226-227.
[32] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives, Theory and Applications Gordon and Breach New York, 1993. 
[33] S.K. Sahoo, M. Tariq, H. Ahmad, J. Nasir, H. Aydi and A. Mukheimer, New Ostrowski-type fractional integral inequalities via generalized exponential type convex functions and applications, Symmetry., 13 (2021), art: 1429.
[34] S.K. Sahoo, P.O. Mohammed, B. Kodamasingh, M. Tariq and Y.S. Hamed, New fractional integral inequalities for convex functions pertaining to Caputo--Fabrizio operator, Frac Fract., 6 (2022), article: 171. 
[35] E. Set, New inequalities of Ostrowski-type for mappings whose derivatives are $s-$convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), pp. 1147-1154.
[36] H.M. Srivastava, S.K. Sahoo, P.O. Mohammed, B. Kodamasingh and Y.S. Hamed, New Riemann--Liouville fractional order inclusions for convex functions via interval valued settings associated with pseudo order relations, Frac Fract., 6 (2022), art: 212.
[37] S. Varosanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), pp. 303-311.
[38] X. Yang, A note on Holder inequality, Appl. Math. Comput., 134 (2003), pp. 319-322.