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JOURNALS // Problemy Fiziki, Matematiki i Tekhniki (Problems of Physics, Mathematics and Technics) // Archive

PFMT, 2020 Issue 3(44), Pages 61–66 (Mi pfmt729)

MATHEMATICS

$\mathfrak{H}_p\mathfrak{H}_q$-convex functions and generalization of the Hölder, Minkowski, and Muirhead inequalities

S. M. Gorskya, V. I. Murashkab

a Saint Petersburg Academic University
b F. Scorina Gomel State University

Abstract: Let $\mathfrak{M}$, $\mathfrak{N}$ be any means. Let $\mathfrak{H}_p$ be a power mean with exponent $p$. A function $f$ is called $\mathfrak{MN}$-convex if for any $x$ and $y$ from the domain of $f$ the inequality$f(\mathfrak{M}(x,y))\leqslant\mathfrak{N}(f(x),f(y))$ holds. In this paper the method of constructing $\mathfrak{H}_p\mathfrak{H}_q$-convex functions is proposed. For such functions generalizations of Cauchy–Schwarz, Hölder, Minkowski, Mahler, and Muirhead inequalities are obtained.

Keywords: $\mathfrak{MN}$-convex function, Cauchy–Schwarz inequality, Hölder inequality, Minkowski inequality, Mahler inequality, Muirhead inequality, Hölder mean.

UDC: 517.162

Received: 11.02.2020



© Steklov Math. Inst. of RAS, 2025