Popoviciu type inequalities for \(n\)-convex functions via extension of weighted Montgomery identity

Asif R. Khan, Hira Nabi, Josip E. Pečarić

Abstract


In this article, we derive the Popoviciu-type inequalities by using the weighted version of the extension of Montgomery's identity and Green functions. By considering the \(n\)-convex function, we prove some identities and related inequalities involving sums \(\sum_{i=1}^\gamma \varrho_{i} \zeta(\chi_{i})\) and integrals \( \int_{\alpha_{1}}^{\beta_{1}} \varrho(\chi) \zeta(g(\chi)) \, d\chi\). Some results for \(n\)-convex functions at a point are also obtained. Besides that, some Ostrowski-type inequalities also present which are interrelated with the obtained inequalities.


Keywords


n-convex functions, n-convex functions at a point, Weighted Montgomery identity, Green's function, Ostrowski type inequalities

Full Text:

PDF

References


Aljinovi c, A.A., Pe cari c, J.E., Vukelic, A., On some Ostrowski type inequalities via Montgomery identity and Taylor's formula II, Tamkang Jour. Math., 36(2005), no. 4, 279-301.

Baloch, I.A., Pe cari c, J.E., Praljak, M., Generalization of Levinson's inequality, J. Math. Inequal., 9(2015), no. 2, 571-586.

Butt, S.I., Mehmood, N., Pe cari c, J.E., New generalization of Popoviciu type inequalities via new Green functions and Taylor's formula, Submitted.

Butt, S.I., Mehmood, N., Pecaric, J.E., New generalizations of Popoviciu type inequalities via new Green functions and Fink's identity, Trans. A. Razmadze Math. Inst., 171(2017), no. 3, 293-303.

Cerone, P., Dragomir, S.S., Some new Ostrowski-type bounds for the Ceby sev functional and applications, J. Math. Inequal., 8(2014), no. 1, 159-170.

Khan, A.R., Latif, N., Pecaric, J.E., Exponential convexity for majorization, J. Inequal. Appl., 2012(2012), no. 1, 1-13.

Khan, A.R., Pecaric, J.E., Positivity of sums and integrals for n-Convex functions via extension of Montgomery identity using new Green functions, Trans. Math. Comp. Sci., 1(2021), no. 2, 51-67.

Khan, A.R., Pecaric, J.E., Praljak, M., Popoviciu type inequalities for n-convex functions via extension of Montgomery identity, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat., 24(2016), no. 3, 161-188.

Khan, A.R., Pe cari c, J.E., Praljak, M., Weighted averages of n-convex functions via

extension of Montgomery's identity, Arab. J. Math., 9(2020), no. 2, 381-392.

Khan, A.R., Pecaric, J.E., Praljak, M., Varo sanec, S., General Linear Inequalities and Positivity/ Higher order convexity, J. Math. Inequal., Monographs in Inequalities, 9, Element, Zagreb 2017, pp. 269.

Khan, A.R., Pe car c, J.E., Varo sanec, S., Popoviciu type characterization of positivity of sums and integrals for convex functions of higher order, J. Math. Inequal., 7(2013), no. 2, 195-212.

Mehmood, N., Agarwal, R.P., Butt, S.I., Pecaric, J.E., New generalizations of Popoviciu-type inequalities via new Green's functions and Montgomery identity, J. Inequal. Appl., 2017(2017), no. 1, 1-17.

Mitrinovi c, D.S., Pecaric, J.E., Fink, A.M., Inequalities for Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.

Pecaric, J.E., On the Cebysev inequality, Bul. Inst. Politehn. Timi soara, 25(1980), no. 39, 5-9.

Pecaric, J.E., On Jessen's inequality for convex functions, III, J. Math. Anal. Appl., 156(1991), 231-239.

Pecaric, J.E., Peru si c, A., Smoljak, K., Generalizations of Ste ensen's inequality by Abel-Gontscharo polynomial, Khayyam J. Math., 1(2015), no. 1, 45-61.

Pecaric, J.E., Praljak, M., Witkowski, A., Linear operators inequality for n{convex functions at a point, Math. Inequal. Appl., 2015(2015), no. 18, 1201-1217.

Pecaric, J.E., Proschan, F., Tong, Y.L., Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, New york, 1992.

Popoviciu, T., Notes sur les fonctions convexes d'ordre superieur III, Mathematica (Cluj), 16(1940), 74-86.

Popoviciu, T., Notes sur les fonctions convexes d'ordre superieur IV, Disqusitiones Math., 1(1940), 163-171.

Popoviciu, T., Notes sur les fonctions convexes d'ordre superieur IX, Bull. Math. Soc. Roumaine Sci., 43(1941), no. 1/2, 85-141.

Popoviciu, T., Les Fonctions Convexes, Hermann and Cie Editeurs, Paris 1944.




DOI: http://dx.doi.org/10.24193/subbmath.2024.1.02

Refbacks

  • There are currently no refbacks.