Browsing by Author "Merentes, Nelson"

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Access status: Open Access ,
Characterizations and decomposition of strongly Wright-convex functions of higher order
(2015) Gilányi, Attila; Merentes, Nelson; Nikodem, Kazimierz; Páles, Zsolt
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 $f$ is strongly Wright-convex of order $n$ if and only if it is of the form $f(x)=g(x)+p(x)+c x^{n+1}$, 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.
Access status: Open Access ,
Integral representation of functions of bounded second Φ-variation in the sense of Schramm
(2012) Giménez, José; Merentes, Nelson; Rivas, Sergio
In this article we introduce the concept of second $\Phi$-variation in the sense of Schramm for normed-space valued functions defined on an interval $[a,b] \subset \mathbb{R}$. To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of $\varphi$-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function $f \in \mathbb{X}^{[a,b]}$ (where $\mathbb{X}$ is a reflexive Banach space) is of bounded second $\Phi$-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm.