Geodesic
$\mathfrak{K}_{\varphi}\mathfrak{K}_{\psi}$-convex functions and generalizations of classical inequalities
Problemy fiziki, matematiki i tehniki, no. 4 (2018), pp. 98-102 Cet article a éte moissonné depuis la source Math-Net.Ru

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A function $f$ is called $\mathfrak{MN}$-convex, if for any $x$ and $y$ from the domain of $f$ inequality $f(\mathfrak{M}(x,y))\leqslant\mathfrak{N}(f(x),f(y))$ holds, where $\mathfrak{M}$ and $\mathfrak{N}$ are means. In this paper geometric interpretation of $\mathfrak{MN}$-convexity of a function is obtained, where $\mathfrak{M}$ and $\mathfrak{N}$ are Kolmogorov's means. For such functions analogies of rearrangement, Popovicu's, Chebyshev's sum and Hermite–Hadamar's inequalities are obtained.
Keywords: convex function, $\mathfrak{MN}$-convex function, rearrangement inequality, Popovicu's inequality, Chebyshev's sum inequality, Jensen's inequality, Hermite–Hadamar's inequality. 
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V. I. Murashka; S. M. Gorsky; Ya. I. Sandryhaila. $\mathfrak{K}_{\varphi}\mathfrak{K}_{\psi}$-convex functions and generalizations of classical inequalities. Problemy fiziki, matematiki i tehniki, no. 4 (2018), pp. 98-102. http://geodesic.mathdoc.fr/item/PFMT_2018_4_a16/

[1] W.W. Breckner, “Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen”, Publ. Inst. Math., 23 (1978), 13–20 | MR | Zbl

[2] E.K. Godunova, V.I. Levin, “Neravenstva dlya funktsii shirokogo klassa, soderzhaschego vypuklye, monotonnye i nekotorye drugie vidy funktsii”, Vychisl. mat. i mat. fiz., Mezhvuz. sb. nauch. trudov, MGPI, M., 1985, 138–142

[3] C.E.M. Pearce, A.M. Rubinov, “P-functions, quasi-convex functions and Hadamard-type inequalities”, J. Math. Anal. Appl., 240 (1999), 92–104 | DOI | MR | Zbl

[4] S. Varosanec, “On $h$-convexity”, J. Math. Anal. Appl., 326 (2007), 303–311 | DOI | MR | Zbl

[5] C.P. Niculescu, “Convexity according to means”, J. Math. Ineq. Appl., 6:4 (2003), 571–579 | MR | Zbl

[6] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen, “Generalized convexity and inequalities”, J. Math. Anal. Appl., 335 (2007), 1294–1308 | DOI | MR | Zbl

[7] A. Baricz, “Geometrically concave univariate distributions”, J. Math. Anal. Appl., 363 (2010), 182–196 | DOI | MR | Zbl

[8] J. Aczèl, “A generalization of the notion of convex functions”, Norske Vid. Selsk. Forhd., Trondhjem, 19:24 (1947), 87–90 | MR | Zbl

[9] M.V. Mihai, F. Mitroi-Symeonidis, “New Extensions of Popoviciu's Inequality”, Mediterr. J. Math., 2015 | DOI | MR

[10] M.W. Alomari, Some properties of $h$-$\mathfrak{MN}$-convexity and Jensen's type inequalities, 2017 | DOI

[11] I. Işcan, “Hermite-Hadamar type inequalities for harmonically convex functions”, Haccetepe J. Math. Stat., 43:6 (2014), 935–942 | MR

[12] M.A. Noor, K.I. Noor, M.U. Awan, “Some characterizations of harmonically log-convex functions”, Proc. Jangeon Math. Soc., 17:1 (2014), 51–61 | MR | Zbl

[13] I. Işcan, “Hermite-Hadamar type inequalities for GA-s-convex functions”, Le Matematiche, LXIX (2014), 129–146 | MR

[14] D. Grinberg, Generalizations of Popoviciu's inequality, 20 March 2008, arXiv: 0803.2958v1 | MR

[15] V. Cirtoaje, “Two generalizations of Popoviciu's inequality”, Crux Mathematicorum, 31:5 (2001), 313–318

[16] L.V. Radzivilovskii, “Obobschenie perestanovochnogo neravenstva i mongolskogo neravenstva”, Matem. prosv. Seriya. 3, 10 (2006), 210–224