Generalization of Favard's and Berwald's Inequalities for Strongly Convex Functions

Authors

  • Muhammad Adil Khan College of Science, Hunan City University, Yiyang 413000
  • Syed Zaheer Ullah Department of Mathematics, University of Peshawar, Peshawar 25000
  • Yuming Chu Department of Mathematics, Huzhou University, Huzhou 313000

DOI:

https://doi.org/10.26713/cma.v10i4.1210

Keywords:

Majorization theorem, Strongly convex function, Favard's inequality, Berwald's inequality

Abstract

In this paper, we give generalization of discrete weighted Favard's and Berwald's inequalities for strongly convex functions. The obtained results are the improvement and generalization of the earlier results.

Downloads

Download data is not yet available.

References

S. Abramovich, Convexity, subadditivity and generalized Jensen's inequality, Ann. Funct. Anal. 4 (2013), 183 – 194, available online at https://projecteuclid.org/euclid.afa/1399899535.

G. A. Anastassiou, Basic and s-convexity Ostrowski and Grüss type inequalities involving several functions, Commun. Appl. Anal. 17 (2013), 189 – 212, available online at http://rgmia.org/papers/v16/v16a83.pdf.

M. Adil Khan, Y.-M. Chu, T.U. Khan and J. Khan, Some new inequalities of Hermite-Hadamard type for s-convex functions with applications, Open Math. 15 (2017), 1414 – 1430, DOI: 10.1515/math-2017-0121.

M. Adil Khan, S. Khalid and J. Peˇcari´c, Refinements of some majorization type inequalities, J. Math. Inequal. 7(1) (2013), 73 – 92, DOI: 10.7153/jmi-07-07.

M. Adil Khan, J. Khan and J. Peˇcari´c, Generalization of Jensen's and Jensen-Steffensen's inequalities by generalized majorization theorem, J. Math. Inequal. 11(4) (2017), 1049 – 1074, DOI: 10.7153/jmi-2017-11-80.

M. Adil Khan, N. Latif, J. Peˇcari´c and I. Peri´c, On Majorization for Matrices, Mathematica Balkanica. 27 Fasc. 1-2 (2013), 19 pages, available online at http://sci-gems.math.bas.bg:8080/jspui/handle/10525/2606.

M. Adil Khan, S. Zaheer Ullah and Y.-M. Chu, The concept of coordinate strongly convex functions and related inequalities, Rev. R. Acad. Cienc. Exactas Fí­s. Nat. Ser. A Mat. RACSAM 113 (2018), 17 pages, DOI: 10.1007/s13398-018-0615-8.

S.-H. Wu, M. Adil Khan, A. Basir and R. Saadati, Some majorization integral inequalities for functions defined on rectangles, J. Inequal. Appl. 2018 (2018), Article ID 146, 13 pages, DOI: 10.1186/s13660-018-1739-2.

M. Adil Khan, S.-H. Wu, H. Ullah and Y.-M. Chu, Discrete majorization type inequaliites for convex functions on rectangles, J. Inequal. Appl. 2019 (2019), Article ID 16, 18 pages, DOI: 10.1186/s13660-019-1964-3.

Y.-M. Chu, M. Adil Khan, T. Ali and S. S. Dragomir, Inequalities for ®-fractional differentiable functions, J. Inequal. Appl. 2017 (2017), Article ID 93, 12 pages, DOI: 10.1186/s13660-017-1371-6.

Y.-M. Chu, M. Adil Khan, T. U. Khan and T. Ali, Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl. 9(6) (2016), 4305 – 4316, available online at http://www.tjnsa.com.

Y.-M. Chu, M.-K. Wang, S.-L. Qiu and Y.-P. Jiang, Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl. 63(7) (2012), 1177 – 1184, DOI: 10.1016/j.camwa.2011.12.038.

Y.-M. Chu and M.-K. Wang, Optimal Lehmer mean bounds for the Toader mean, Results Math. 61(3-4) (2012), 223 – 229, DOI: 10.1007/s00025-010-0090-9.

Y.-M. Chu, G.-D. Wang and X.-H. Zhang, Schur convexity and Hadamard's inequality, Math. Inequal. Appl. 13(4) (2010), 725 – 731, DOI: 10.7153/mia-13-51.

Y.-M. Chu, G.-D. Wang and X.-H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function, Math. Nachr. 284(5-6) (2011), 653 – 663, DOI: 10.1155/2018/4036942.

Y.-M. Chu, W.-F. Xia and X.-H. Zhang, The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications, J. Multivariate Anal. 105 (2012), 412 – 421, DOI: 10.1016/j.jmva.2011.08.004.

B. Definetti, Sulla stratificazioni convesse, Ann. Math. Pura. Appl. 30 (1949), 173 – 183, DOI: 10.1007/BF02415006.

S.S. Dragomir, Some majorization type discrete inequalities for convex functions, Math. Ineq. Appl. 7(2) (2004), 207 – 216, available online at https://rgmia.org/papers/v6e/MTICF.pdf.

S.S. Dragomir, Inequalities of Hermite-Hadamard type for '-convex functions, Preprint RGMIA Res. Rep. Coll. 14(16) (2013), Article ID 87, available online at http://rgmia.org/papers/v16/v16a87.pdf.

S.S. Dragomir, Inequalities of Hermite-Hadamard type for ¸-convex functions on linear spaces, Preprint RGMIA Res. Rep. Coll. 18(17) (2014), Article ID 13, available online at https://rgmia.org/papers/v17/v17a13.pdf.

L. Fuchs, A new proof of an inequality of Hardy-Littlewood-Pólya, Mat. Tidssker. 13 (1947), 53 – 54. [22] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd edition, Cambridge University Press, England (1952).

D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821 – 828.

J. L. W. V. Jensen, On konvexe funktioner og uligheder mellem middlvaerdier, Nyt. Tidsskr. Math. B. 16 (1905), 49 – 69.

J. Khan, M. Adil Khan and J. Peˇcari´c, On Jensen's Type Inequalities via Generalized Majorization Inequalities, Filomat 32(16) (2018), 5719 – 5733, DOI: 10.2298/FIL1816719K.

Y. Khurshid, M. Adil Khan, Y.-M. Chu, Z. A. Khan and L.-S. Liu, Hermite-Hadamard-Fejér inequalities for conformable fractional integrals via preinvex functions, J. Funct. Spaces. 2019 (2019), Article ID 3146210, 9 pages, DOI: 1155/2019/3146210.

J. Karamata, Sur une inégalité relative aux functions convexes, Pub. Math. Univ. Belgrade 7 (1932), 145 – 148.

N. Latif, J. Peˇcari´c and I. Peri´c, On discrete Favard's and Berwald's inequalities, Commun. Math. Anal. 12(2) (2012), 34 – 57, available online at https://projecteuclid.org/euclid.cma/1331929870.

O. L. Mangasarian, Pseudo-convex functions, SIAM. Journal on Control. 3 (1965), 281 – 290, DOI: 10.1016/B978-0-12-780850-5.50009-5.

D. S. Mitrinovi´c, Analytic Inequalities, Springer-Verlag, Berlin and New York (1970), available online at https://sites.math.washington.edu/~morrow/334-16/kazarinoff.pdf.

N. Merentes and K. Nikodem, Remarks on strongly convex functions, Aequ. Math. 80 (2010), 193 – 199, DOI: 10.1007/s00010-010-0043-0.

A. W. Marshall and I. Olkin, Inequalities, Theory of Majorization and its Applications, Academic Press, New York (1979), available online at https://www.springer.com/gp/book/9780387400877.

H. R. Moradi, M. E. Omidvar, M. Adil Khan and K. Nikodem, Around Jensen's inequality for strongly convex functions, Aequ. Math. 92 (2018), 25 – 37, DOI: 10.1007/s00010-017-0496-5.

M. E. í–zdemir, M. Avci and H. Kavurmaci, Hermite-Hadamard-type inequalities via ((alpha,m))-convexity, Comput. Math. Appl. 61 (2011), 2614 – 2620, DOI: 10.1016/j.camwa.2011.02.053.

J. Peˇcari´c, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, New York (1992), available online at https://www.elsevier.com/.../convex-functions-partial-orderings-and-statistical-applications.pdf.

T. Rajba, On strong delta-convexity and Hermite-Hadamard type inequalities for delta-convex functions of higher order, Math. Inequal. Appl. 18 (2015), 267 – 293, DOI: 10.7153/mia-18-20.

Y.-Q. Song, M. Adil Khan, S. Zaheer Ullah and Y.-M. Chu, Integral inequalities involving strongly convex functions, J. Funct. Spaces. 2018 (2018), Article ID 6596921, 8 pages, DOI: 10.1155/2018/6595921.

E. Set, M. E. í–zdemir and S. S. Dragomir, On Hadamard-type inequalities involving several kinds of convexity, J. Inequal. Appl. 12 (2010), Art. ID 286845, 12 pages, DOI: 10.1155/2010/286845.

S. Varošanec, On h-convexity, J. Math. Anal. Appl. 326 (2007), 303 – 311, DOI: 10.1016/j.jmaa.2006.02.086.

M.-K. Wang and Y.-M. Chu, Landen inequalities for a class of hypergeometric functions with applications, Math. Inequal. Appl. 21(2) (2018), 521 – 537, DOI: 10.7153/mia-2018-21-38.

M.-K. Wang, Y.-M. Li and Y.-M. Chu, Inequalities and infinite product formula for Ramanujan generalized modular equation function, Ramanujan J. 46(1) (2018), 189 – 200, DOI: 10.1007/s11139-017-9888-3.

Z.-H. Yang, Y.-M. Chu and W. Zhang, High accuracy asymptotic bounds for the complete elliptic integral of the second kind, Appl. Math. Comput. 348 (2019), 552 – 564, DOI: 10.1016/j.amc.2018.12.025.

Z.-H. Yang, W.-M. Qian and Y.-M. Chu, Monotonicity properties and bounds involving the complete elliptic integrals of the first kind, Math. Inequal. Appl. 21(4) (2018), 1185 – 1199, DOI: 10.7153/mia-2018-21-82.

Z.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the error function, Math. Inequal. Appl. 21(2) (2018), 469 – 479, DOI: 10.7153/mia-2018-21-32.

S. Zaheer Ullah, M. Adil Khan and Y.-M. Chu, Majorization theorems for strongly convex functions, J. Inequal. Appl. 2019 (2019), 13 pages, DOI: 10.1186/s13660-019-2007-9.

X.-M. Zhang, Y.-M. Chu and X.-H. Zhang, The Hermite-Hadamard type inequality of GA-convex functions and its applications, J. Inequal. Appl. 2010 (2010), Article ID 507560, 11 pages, DOI: 10.1155/2010/507560.

Downloads

Published

31-12-2019
CITATION

How to Cite

Khan, M. A., Ullah, S. Z., & Chu, Y. (2019). Generalization of Favard’s and Berwald’s Inequalities for Strongly Convex Functions. Communications in Mathematics and Applications, 10(4), 693–705. https://doi.org/10.26713/cma.v10i4.1210

Issue

Section

Research Article