$\mathfrak{K}_{\varphi}\mathfrak{K}_{\psi}$-convex functions and generalizations of classical inequalities

V. I. Murashka; S. M. Gorsky; Ya. I. Sandryhaila

Geodesic

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$\mathfrak{K}_{\varphi}\mathfrak{K}_{\psi}$-convex functions and generalizations of classical inequalities

V. I. Murashka ; S. M. Gorsky ; Ya. I. Sandryhaila

Problemy fiziki, matematiki i tehniki, no. 4 (2018), pp. 98-102

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Résumé

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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     author = {V. I. Murashka and S. M. Gorsky and Ya. I. Sandryhaila},
     title = {$\mathfrak{K}_{\varphi}\mathfrak{K}_{\psi}$-convex functions and generalizations of classical inequalities},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {98--102},
     publisher = {mathdoc},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2018_4_a16/}
}

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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/