New proofs of Fejer's and discrete Hermite-Hadamard inequalities with applications

Yıl 2023, Cilt: 72 Sayı: 4, 1110 - 1125, 29.12.2023

Çağla Sekin , Mehmet Emin Tamar , İlham Aliyev

https://doi.org/10.31801/cfsuasmas.1262668

Öz

New proofs of the classical Fejer inequality and discrete Hermite-Hadamard inequality (HH) are presented and several applications are given, including (HH)-type inequalities for the functions, whose derivatives have inflection points. Morever, some estimates from below and above for the first moments of functions $f:[a,b]\rightarrow \mathbb{R}$ about the midpoint $c=(a+b)/2$ are obtained and the reverse Hardy inequality for convex functions $f:(0,\infty )\rightarrow (0,\infty )$ is established.

Anahtar Kelimeler

Fejer inequality, convex functions, discrete Hermite-Hadamard inequality, Jensen inequality, Hardy inequality

Kaynakça

• Amrahov, S. E., A note on Hadamard inequalities for the product of the convex functions, International J. of Research and Reviews in Applied Sciences, 5(2) (2010), 168-170.
• Alomari, M., Darus, M. and Dragomir, S. S., New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex, Tamkang J. Math., 41(4) (2010), 353-359.
• Azpetia, A. G., Convex functions and the Hadamard inequality, Revista Colombiana Mat., 28 (1994), 7-12.
• Bakula, M. K., Özdemir, M. E. and Pecaric, J., Hadamard-type inequalities for m-convex and $(\alpha,m)$-convex functions. J. Ineq. Pure Appl. Math., 9(4) (2008), Article 96.
• Chen, F., Liu, X., On Hermite-Hadamard type inequalities for functions whose second derivatives absolute values are s-convex, ISRN Applied Mathematics, (2014), 1-4, DOI:10.1155/2014/829158.
• Dragomir, S. S., Pearce, C. E. M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
• Dragomir, S. S., On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 3(1) (2002).
• El Farissi, A., Simple proof and refinement of Hermite-Hadamard inequality. J. Math. Inequal, 4(3) (2010), 365-369.
• Fink, A. M., A best possible Hadamard inequality, Math. Inequal Appl., 1 (1998), 223-230.
• Florea, A., Niculescu, C. P., A Hermite-Hadamard inequality for convex-concave symmetric functions, Bull. Soc. Sci. Math. Roum., 50(98) No:2 (2007), 149-156.
• Hwang, D. Y., Tseng, K. L. and Yang, G. S., Some Hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane, Taiwanese J. Math., 11(1) (2007), 63-73, DOI:10.11650/twjm/1500404635.
• Ion, D. A., On an inequality due to Amrahov, Annals of University of Craiova, Math. Comp. Sci. Ser., 38(1) (2011), 92-95, DOI 10.52846/ami.v38i1.396.
• Kemali, S., Yesilce, I., Adilov, G., B-Convexity, B-1-Convexity and Their Comparison, Numerical Functional Analysis and Optimization, 36(2) (2015), 133-146,. DOI:10.1080/01630563.2014.970641.
• Kemali, S., Sezer, S., Tınaztepe, G., Adilov, G., s-Convex functions in the third sense, Korean Journal of Mathematics, 29(3) (2021), 593-602, DOI 10.11568/kjm.2021.29.3.593.
• Mercer, A. M. D., A variant of Jensen’s inequality, J. Ineq. Pure and Appl. Math., 4(4) (2003), Article 73.
• Mitrinovic, D. S., Pecaric, J. E. and Fink, A. M., Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.
• Niculescu C. P., Persson, L. E., Old and new on the Hermite-Hadamard inequality, Real Anal. Exchange, 29 (2003/2004), 663-686.
• Niculescu C. P., Persson, L. E., Convex Functions and Their Applications, A Contemporary Approach, CMS Books in Mathematics, V.23, Springer-Verlag, 2006.
• Sarikaya, M. Z., Saglam, A. and Yildirim, H., New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex or quasi-convex, International J. of Open Problems in Comp. Sci. and Math., 5(3) (2012), 1-14.
• Sezer, S., Eken, Z., Tınaztepe, G., Adilov, G., p-Convex functions and some of their properties,Numerical Functional Analysis and Optimization, 42(4) (2021), 443-459, DOI:10.1080/01630563.2021.1884876.
• Tseng, K. L., Hwang S. R. and Dragomir, S. S., On some new inequalities of Hermite-Hadamard-Fejer type involving convex functions, Demons. Math., 40(1) (2007), 51-64, DOI:10.1515/dema-2007-0108.
• Qi, F., Yang, Z. L., Generalizations and refinements of Hermite-Hadamard’s inequality, Rocky Mountain J. Math., 35 (2005), 235-251.