The concept of coordinate strongly convex functions and related inequalities
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In this paper, we introduce a new class of functions known as coordinate strongly convex functions. We discuss the relation between strongly convex functions and coordinate strongly convex functions. Also, we present some natural properties of coordinate strongly convex functions. We present Slater’s, Jensen’s and converse of the Jensen inequalities in discrete as well as integral versions for coordinate strongly convex functions. Furthermore, we present Hermite–Hadamard’s type inequalities for coordinate strongly convex functions.
KeywordsCoordinate strongly convex function Jensen’s inequality Slater’s inequality
Mathematics Subject ClassificationPrimary 26D15 26A51 Secondary 39B62
The research was supported by the Natural Science Foundation of China (Grant Nos. 61373169, 11701176, 11601485 ) and the Science and Technology Research Program of Zhejiang Educational Committee (Grant no. Y201635325). The third author Yu-Ming Chu is the corresponding author of the article. The authors would like to express their sincere thanks to anonymous referees for their valuable suggestions and comments which helped the authors to improve this article substantially.
- 5.Adil Khan, M., Khan, G. A., Ali, T., Batbold, T., Kiliçman, A.: Further refinement of Jensen’s type inequalities for the function defined on the rectangle, Abstr. Appl. Anal. 1–8 (2013)Google Scholar
- 9.Adil Khan, M., Zaheer Ullah, S., Reza Moradi, Hamid.: Jensen type inequalities for strongly convex functions, (submitted) Google Scholar
- 15.Dragomir, S. S.: Inequalities of Hermite-Hadamard type for \(\varphi \)-convex functions, Preprint RGMIA Res. Rep. Coll. 16, 14, Art. 87, (2013). http://rgmia.org/papers/v16/v16a87.pdf
- 16.Dragomir, S. S.: Inequalities of Hermite–Hadamard type for \(\lambda \)-convex functions on linear spaces, Preprint RGMIA Res. Rep. Coll. 17, 18 Art. 13, (2014). http://rgmia.org/papers/v17/v17a13.pdf
- 22.Moradi, H. R., Omidvar, M. E., Adil Khan, M., Nikodem, K.: Around Jensen’s inequality for strongly convex functions, Aequat. Math. (2017). https://doi.org/10.1007/s00010-017-0496-5
- 25.Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966)Google Scholar
- 27.Song, Y-Qing., Adil Khan, M., Zaheer Ullah, S., Chu, Yu-Ming.: Integral inequalities involving strongly convex functions, Journal of Function Spaces. 2018 (2018) Article ID 6595921Google Scholar
- 30.Set, E., Özdemir, M. E., Dragomir, S. S.: On Hadamard-type inequalities involving several kinds of convexity. J. Inequal. Appl. 12 (2010) Art. ID 286845Google Scholar