Jensen's type inequalities for two times differentiable functions with applications

Year 2025, Volume: 74 Issue: 3, 424 - 445, 23.09.2025

Muhammad Adil Khan , Shahid Khan , Dilda Pečarić , Josip Pečarić

https://doi.org/10.31801/cfsuasmas.1519488

Abstract

In the main body, first of all this work recommends an inequality of Jensen's type involving Green functions for a class of two times differentiable functions. This result enables further to obtain some related interpolating inequalities for a function $f$ such that $|f''|^{q}$ is either concave or convex for $q\geq1$. Then manipulation of certain existing results in the corresponding interpolating inequalities gives bounds for the differences of the Jensen-Steffensen and Jensen's inequalities. In the similar fashion, they provide some new variants for the reverse form of aforementioned inequalities. Further, the obtained results about Jensen's inequality yield different novel adaptations of Hölder's inequality, fresh insights into the discrepancy of the well known Hermite-Hadamard inequality, and inequalities for geometric mean, quasi-arithmetic mean, and power mean. As a resultant, this work also suggests graphical interpretation of some results to verify the authenticity and sharpness of the obtained results about Jensen's inequality. Finally, this research work put forward some applications involving Zipf-Mandelbrot entropy and various types of Csiszár divergence from information theory.

Keywords

Jensen inequality , Green function , Jensen-Steffensen inequality , divergences

References

• Adamek, M., On a Jensen-type inequality for F-convex functions, Math. Inequal. Appl., 22(4) (2019), 1355–1364. https://doi.org/10.7153/mia-2019-22-93.
• Adeel, M., Khan, K. A., Pečarić, Ð., Pečarić, J., Levinson type inequalities for higher order convex functions via Abel-Gontscharoff interpolation, Adv. Difference Equ., 2019 (2019). https://doi.org/10.1186/s13662-019-2360-5.
• Adil Khan, M., Ali, T., Din, Q., Kilicman, A., Refinements of Jensen’s inequality for convex functions on the co-ordinates of a rectangle from the plane, Filomat, 30(3) (2016), 803–814.
• Adil Khan, M., Khan, S., Chu, Y.-M., A new bound for the Jensen gap with applications in information theory, IEEE Access, 8 (2020), 98001–98008. https://doi: 10.1109/ACCESS.2020.2997397.
• Adil Khan, M., Khan, S., Pečarić, Ð., Pečarić, J., New improvements of Jensen’s type inequalities via 4-convex functions with applications, RACSAM, 115(2) (2021). https://doi.org/10.1007/s13398-020-00971-8.
• Adil Khan, M., Khan, S., Ullah, I., Khan, K. A., Chu, Y. M., A novel approach to the Jensen gap through Taylor’s theorem, Math. Methods Appl. Sci., 44(5) (2021), 3324-3333. https://doi.org/10.1002/mma.6944.
• Adil Khan, M., Wu, S.-H., Ullah, H., Chu, Y.-M., Discrete majorization type inequalities for convex functions on rectangles, J. Inequal. Appl., 2019(1) (2019), 1-18. https://doi.org/10.1186/s13660-019-1964-3.
• Ahmad, K., Adil Khan, M., Khan, S., Ali, A., Chu, Y.-M., New estimation of Zipf-Mandelbrot and Shannon entropies via refinements of Jensen’s inequality, AIP Adv., 11(1) (2021), Article 015147. https://doi.org/10.1063/5.0039672.
• Aslani, S. M., Delavar, M. R., Vaezpour, S. M., Inequalities of Fejér type related to generalized convex functions with applications, Int. J. Anal. Appl., 16(1) (2018), 38–49. https://doi.org/10.28924/2291-8639-16-2018-38.
• Brudnyi, A., Lq norm inequalities for analytic functions revisited, J. Approx. Theory, 179 (2014), 24–32. https://doi.org/10.1016/j.jat.2013.11.008.
• Chu, H. H., Kalsoom, H., Rashid, S., Idrees, M., Safdar, F., Chu, Y. M., Baleanu, D., Quantum analogs of Ostrowski- type inequalities for Raina’s function correlated with coordinated generalized Φ-convex functions, Symmetry, 12(2) (2020), Article 308. https://doi.org/10.3390/sym12020308.
• Cortez, M. V., Abdeljawad, T., Mohammed, P. O., Oliveros, Y. R., Simpson’s integral inequalities for twice differentiable convex functions, Math. Probl. Eng., 2020 (2020), Article ID 1936461. https://doi.org/10.1155/2020/1936461.
• Farid, G., Khan, K. A., Latif, N., Rehman, A. U., Mehmood, S., General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2018 (2018), Article ID 243. https://doi.org/10.1186/s13660-018-1830-8.
• Iqbal, A., Adil Khan, M., Suleman, M., Chu, Y. M., The right Riemann-Liouville fractional Hermite-Hadamard type inequalities derived from Green’s function, AIP Adv., 10(4) (2020), Article 045032. https://doi.org/10.1063/1.5143908.
• Kalsoom, H., Rashid, S., Idrees, M., Safdar, F., Akram, S., Baleanu, D., Chu, Y. M., Post quantum integral inequalities of Hermite-Hadamard-type associated with co-ordinated higher-order generalized strongly pre-invex and quasi-pre-invex mappings, Symmetry, 12(3) (2020), Article 443. https://doi.org/10.3390/sym12030443.
• Khalid, S., Pečarić, J., On the refinements of the integral Jensen-Steffensen inequality, J. Inequal. Appl., 2013 (2013), Article ID 20. https://doi.org/10.1186/1029-242X-2013-20.
• Khalid, S., Pečarić, J., Refinements of some Hardy-Littlewood Pólya type inequalities via Green’s functions and Fink’s identity and related results, J. Inequal. Appl., 2020 (2020), Article ID 260. https://doi.org/10.1186/s13660-020-02498-3.
• Khan, S., Adil Khan, M., Butt, S. I., Chu, Y. M., A new bound for the Jensen gap pertaining twice differentiable functions with applications, Adv. Difference Equ., 2020 (2020), Article ID 333. https://doi.org/10.1186/s13662-020- 02794-8.
• Khan, S., Adil Khan, M., Chu, Y. M., Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Methods Appl. Sci., 43 (2020), 2577–2587. https://doi.org/10.1002/mma.6066.
• Khan, S., Adil Khan, M., Chu, Y. M., New converses of Jensen inequality via Green functions with applications, RACSAM, 114(3) (2020), Article No. 114. https://doi.org/10.1007/s13398-020-00843-1.
• Khan, M. B., Noor, M. A., Abdullah, L., Chu, Y. M., Some new classes of preinvex fuzzy-interval-valued functions and inequalities, Int. J. Comput. Intell. Syst., 14(1) (2021), 1403–1418. https://DOI: 10.2991/ijcis.d.210409.001.
• Khurshid, Y., Adil Khan, M., Chu, Y. M., Ostrowski type inequalities involving conformable integrals via preinvex functions, AIP Adv., 10(5) (2020), Article 055204. https://doi.org/10.1063/5.0008964.
• Mehmood, N., Butt, S. I., Horváth, L., Pečarić, J., Generalization of cyclic refinements of Jensen’s inequality by Fink’s identity, J. Inequal. Appl., 2018 (2018), Article ID 51. https://doi.org/10.1186/s13660-018-1640-z.
• Mehrez, K., Agarwal, P., New Hermite-Hadamard type integral inequalities for convex functions and their applications, J. Comput. Appl. Math., 350 (2019), 274–285. https://doi.org/10.1016/j.cam.2018.10.022.
• Mikić, R., Pečarić, J., Rodić, M., Levinson’s type generalization of the Jensen inequality and its converse for real Stieltjes measure, J. Inequal. Appl., 2017 (2017), Article ID 4. https://doi.org/10.1186/s13660-016-1274-y.
• Niaz, T., Khan, K. A., Pečarić, J., On refinement of Jensen’s inequality for 3-convex function at a point, Turkish J. Ineq., 4(1) (2020), 70–80.
• Pečarić, J., Proschan, F., Tong, Y. L., Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York: 1992.
• Steffensen, J. F., On certain inequalities and methods of approximation, J. Inst. Actuaries, 51 (1919), 274–297. https://doi:10.1017/S0020268100028687.
• Sun, M. B., Chu, Y. M., Inequalities for the generalized weighted mean values of g-convex functions with applications, RACSAM, 114 (2020), Article No. 172. https://doi.org/10.1007/s13398-020-00908-1.
• Wu, S. H., Adil Khan, M., Haleemzai, H. U., Refinements of majorization inequality involving convex functions via Taylor’s theorem with mean value form of the remainder, Mathematics, 7(8) (2019), Article ID 663. https://doi.org/10.3390/math7080663.