×

Some Hardy-type inequalities for convex functions via delta fractional integrals. (English) Zbl 1495.26032

Summary: In this paper, some Jensen- and Hardy-type inequalities for convex functions are extended by using Riemann-Liouville delta fractional integrals. Further, some Pólya-Knopp-type inequalities and Hardy-Hilbert-type inequality for convex functions are also proved. Moreover, some related inequalities are proved by using special kernels. Particular cases of resulting inequalities provide the results on fractional calculus, time scales calculus, quantum fractional calculus and discrete fractional calculus.

MSC:

26D15 Inequalities for sums, series and integrals
26A33 Fractional derivatives and integrals
26A51 Convexity of real functions in one variable, generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dolbeault, J., Esteban, M. J. and Séré, E., On the eigenvalues of operators with gaps. Application to Dirac operators, J. Funct. Anal.174 (2000) 208-226. · Zbl 0982.47006
[2] Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, (Cambridge University Press, Cambridge, 1988). · Zbl 0634.26008
[3] Hardy, G. H., Notes on a theorem of Hilbert, Math. Z.6 (1920) 314-317. · JFM 47.0207.01
[4] Hardy, G. H., Notes on some points in the integral calculus (LX), An inequality between integrals, Messenger Math.54 (1625) 150-156. · JFM 51.0192.01
[5] Hardy, G. H., Notes on some points in the integral calculus (LXIT), Messenger Math.57 (1928) 12-16. · JFM 53.0217.02
[6] Pachpatte, B. G., A note on Copson’s inequality involving series of positive terms, Tamkang J. Math.21 (1990) 13-19. · Zbl 0705.26016
[7] Saker, S. H., Mahmoud, R. R., Osman, M. M. and Agarwal, R. P., Some new generalized forms of Hardys type inequality on time scales, J. Math. Anal. Appl.20(2) (2017) 459-481. · Zbl 1367.26048
[8] Saker, S. H. and O’Regan, D., Hardy and littlewood inequalities on time scales, Bull. Malaysian Math. Sci. Soc.39 (2016) 527-543. · Zbl 1342.26065
[9] Opic, B. and Kufner, A., Hardy-type Inequalities, (Longman Scientific and Technical, Harlow, 1990). · Zbl 0698.26007
[10] Nguyen, D. T., Lam-Hoang, N. and Nguyen, T. A., Hardy and Rellich inequalities with exact missing terms on homogeneous groups, J. Math. Soc. Jpn.71 (2019) 1243-1256. · Zbl 1442.22012
[11] Duy, N. T., Lam, N., Triet, N. and Yin, E., Improved Hardy inequalities with exact remainder terms, Math. Inequal. Appl.23 (2020) 1205-1226. · Zbl 1454.26022
[12] A. Salort, Hardy inequalities in fractional Orlicz-Sobolev spaces, preprint (2020), arXiv:2009.06431. · Zbl 1493.46049
[13] N. Kutev and T. Rangelov, Hardy inequalities with double singular weights, preprint (2020), arXiv:2001.07368. · Zbl 1463.35392
[14] X. Cabre and P. Miraglio, Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, preprint (2019), arXiv:1912.09282.
[15] Kaijser, S., Persson, L. E. and Öberg, A., On Carleman and Knopp’s inequalities, J. Approx. Theory117 (2002) 140-151. · Zbl 1049.26014
[16] Bohner, M., Nosheen, A., Pečarić, J. and Younus, A., Some dynamic Hardy-type inequalities with general kernel, J. Math. Inequal.8 (2014) 185-199. · Zbl 1294.26020
[17] Karpuz, B., Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients, Electron. J. Qual. Theory Differ. Equ.34 (2009) 1-14. · Zbl 1184.34072
[18] Anwar, M., Bibi, R., Bohner, M. and Pečarić, J., Integral inequalities on time scales via the theory of isotonic linear functionals, Abstr. Appl. Anal. (2011) 483595. · Zbl 1221.26026
[19] Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (Springer Science & Business Media, New York, 2001). · Zbl 0978.39001
[20] Wong, F. H., Yeh, C. C. and Lian, W. C., An extension of Jensens inequality on time scales, Adv. Dyn. Syst. Appl.1 (2006) 113-120. · Zbl 1123.26021
[21] Asliyüce, S. and Güvenilir, A. F., Fractional Jensens inequality, Palestian J. Math.7 (2018) 554-558. · Zbl 1393.26021
[22] Iqbal, S., Krulić, K. and Pečarić, J., On an inequality of HG Hardy, J. Inequal. Appl.2010 (2010) 1-23. · Zbl 1229.26014
[23] Bohner, M. and Guseinov, G. S., Multiple Lebesgue integration on time scales, Adv. Diff. Equ. (2006) 026391. · Zbl 1139.39023
[24] Bohner, M. and Guseinov, G. S., The convolution on time scales, Abstr. Appl. Anal. (2007) 058373. · Zbl 1155.39010
[25] Godunova, E. K., Inequalities based on convex functions, Izv. Vysh. Uchebn. Zaved. Matematika47 (1965) 45-53.
[26] Kaijser, S., Nikolova, L., Persson, L. E. and Wedestig, A., Hardy-type inequalities via convexity, Math. Inequal. Appl.8 (2005) 403-417. · Zbl 1083.26013
[27] Littlewood, J. E. and Hardy, G. H., Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, J. fr die Reine und Angew. Math.157 (1927) 141-158. · JFM 53.0193.03
[28] Machihara, S., Ozawa, T. and Wadade, H., Remarks on the Hardy type inequalities with remainder terms in the framework of equalities, in Asymptotic Analysis for Nonlinear Dispersive and Wave Equations (Mathematical Society of Japan, 2019), pp. 247-258. · Zbl 1435.26017
[29] Mironescu, P., The role of the Hardy type inequalities in the theory of function spaces, Rev. Roum. Mathm. Pures Appl.63 (2018) 447-525. · Zbl 1424.46050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.