On Reverse and Generalized of Bellman Type Inequality

Year 2024, Volume: 24 Issue: 3, 567 - 573, 27.06.2024

Necmettin Alp

https://doi.org/10.35414/akufemubid.1373603

Abstract

Bu çalışmanın amacı, önceki çalışmalar incelendikten sonra dışbükey ve integrallenebilir ve diferansiyellenebilir fonksiyonlar için ters Bellman tipi eşitsizliği elde etmektir. Bununla birlikte Bellman tipi eşitsizliğin genelleştirilmesinin ispatı yapılmıştır. Elde edilen ana sonuçlar yardımıyla dışbükey ve integrallenebilir, diferansiyellenebilir fonksiyonlar üzerine bazı özel durumlar elde edildi. Bu sonuçlar kullanılarak analitik olarak hesaplanması zor integraller için bir alt sınır elde dilmiştir

Keywords

konvekslik, eşitsizlikler, integral operatörü, bellman

On Reverse and Generalized of Bellman Type Inequality

Abstract

The aim of this work, to obtain reverse Bellman type inequality for convex and integrable-differentiable functions after examining the previous studies on the Bellman inequality and giving the results that are a source of inspiration for us. With that we proved the generalized of Bellman type
inequality. With the help of obtained main results, we have obtained some special cases on convex and integrable-differentiable functions. For convexity we have also included some examples to make these exercises more understandable. On the other hand, we have expressed with examples
to obtain a lower bound for hard integrals.

Keywords

convexity, inequalities, integral operator, bellman

References

• Alp, N., 2021. Wirtinger type q-Integral Inequalities On q-Calculus. Math. Meth. Appl. Sci., 1–12. https://doi.org/10.1002/mma.7643
• Bellman, R.,1959. On inequalities with alternating signs. Proc. Amer. Math. Soc. 10, 807-809.
• Cristescu, G. And Lupsa, L., 2002. Non-connected Convexities and Applications. Kluwer Academic Publishers, Dordrecht.
• Delgado, J. G. G., Valdes, J. E. N. and Reyes, E. P., 2021. Several integral inequalities for generalized Riemann-Liouville fractional operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70.1: 269-278. https://doi.org/10.31801/cfsuasmas.771172
• Godunova, E. K. and Levin V. I., 1968. A general class of inequalities containing Steffensen’s inequality. Mat. Zametki 3, 339-344.
• Iddrisu, M. M., Okpoti, C. A., and Gbolagade, K. A., 2014. Geometrical proof of new Steffensen’s inequality and applications. Adv. Inequal. Appl., 23, 2050-7461.
• Mitrinovi’c, D. S., and Peˇcari´c, J. E., 1988. On the Bellman generalization of Steffensen’s inequality. III. Journal of mathematical analysis and applications 135.1, 342-345. https://doi.org/10.1016/0022-247X(88)90158-8
• Mirzapour, F., Morassaei, A. and Moslehian, M.S., 2014. More on operator Bellman inequality. Quaest. Math., 37, 9–17. https://doi.org/10.2989/16073606.2013.779965
• Morassaei, A., Mirzapour, F. and Moslehian, M.S., 2013. Bellman inequality for Hilbert space operators. Linear Algebra Appl., 438, 3776–3780. https://doi.org/10.1016/j.laa.2011.06.042
• Pecaric, J. E., 1982. On the Bellman Generalization of Steffensen’s Inequality, J. Math. Anal. Appl. 88, 505-507. https://doi.org/10.1016/0022-247X(82)90208-6
• Roberts, A.W. and Varberg, D.E., 1973. Convex Functions, Academic Press, 300pp, New York.
• Sababheh, M., Moradi, H. R. and Furuichi, S., 2020. Reversing Bellman operator inequality. J. Math. Inequal, 14.2: 577-584. https://doi.org/10.7153/jmi-2020-14-36
• Steffensen, J., 1947. Bounds of certain trigonometric integrals. Tenth Scandinavian Mathematical congress, 181-186, Copenhagan, Hellerup.
• Vivas-Cortez, M., Abdeljawad, T. and Mohammed, P. O., 2020. Simpson’s integral inequalities for twice differentiable convex functions. Math. Probl. Eng., 1936461,1-15. https://doi.org/10.1155/2020/1936461
• Youness, E. A., 1999. E-convex sets, E-convex functions, and E-convex programming. J. Optim. Theory Appl., 102, 439–450. https://doi.org/10.1023/A:1021792726715