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Characterizations and decomposition of strongly wright-convex functions of higher order

Gilanyi A., Merentes N., Nikodem K., Pales Z.

Opuscula Mathematica

2015

Vol. 35, no. 1

37--46

Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661-665] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function / is strongly Wright-convex of order n if and only if it is of the form [formula], where g is a (continuous) n-convex function and p is a polynomial function of degree n. This is a counterpart of Ng's decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives.

Hermite-Hadamard inequalities for convex set-valued functions

Mitroi F. C., Nikodem K., Wąsowicz S.

Demonstratio Mathematica

2013

Vol. 46, nr 4

655--662

The following version of the weighted Hermite-Hadamard inequalities for set-valued functions is presented: Let Y be a Banach space and F : [a, b]→cl(Y) be a continuous set-valued function. If F is convex, then (…) , where μ is a Borel measure on [a, b] and xμ is the barycenter of on [a, b]. The converse result is also given.

On strongly midconvex functions

Azocar A., Gimenez J., Nikodem K., Sanchez J. L.

Opuscula Mathematica

2011

Vol. 31, no. 1

15-26

In this paper we collect some properties of strongly midconvex functions. First, counterparts of the classical theorems of Bernstein-Doetsch, Ostrowski and Sierpiński are presented. A version of Rod é support theorem for strongly midconvex functions and a Kuhn-type result on the relation between strongly midconvex functions and strongly t-convex functions are obtained. Finally, a connection between strong midconvexity and generalized convexity in the sense of Beckenbach is established