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# On conditionally δ-convex functions

Acta Mathematica Hungarica
Authors: Adam Najdecki, Jacek Tabor, and Józef Tabor

## Abstract

Let X be a real vector space, V a subset of X and δ ≧ 0 a given number. We say that f: V → ℝ is a conditionally δ-convex function if for each convex combination t 1 υ 1 + … + t n υ n of elements of V such that t 1 υ 1 + … + t n υ nV the following inequality holds true:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$f(t_1 v_1 + \cdots + t_n v_n ) \leqq t_1 f(v_1 ) + \cdots + t_n f(v_n ) + \delta .$$ \end{document}
We prove that f: V → ℝ is conditionally δ-convex if and only if there exists a convex function
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde f$$ \end{document}
: conv V → [−∞, ∞) such that
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde f(v) \leqq f(v) \leqq \tilde f(v) + \delta for v \in V.$$ \end{document}
In case X = ℝn some conditions equivalent to conditional δ-convexity are also presented.
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# Bounds having Riemann type quantum integrals via strongly convex functions

Studia Scientiarum Mathematicarum Hungarica

The aim of this paper is to obtain some new bounds having Riemann type quantum integrals within the class of strongly convex functions. The results obtained are sharp on limit q → 1. These new results reduce to Tariboon-Ntouyas, Merentes-Nikodem and other previously known results when q → 1, where 0 < q < 1. The sharpness of the results of Tariboon-Ntouyas and Merentes-Nikodem is proved as a consequence.

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# Complete monotonicity of two functions involving the tri-and tetra-gamma functions

Periodica Mathematica Hungarica
Authors: Jiao-Lian Zhao, Bai-Ni Guo, and Feng Qi

## Abstract

The psi function ψ(x) is defined by ψ(x) = Γ′(x)/Γ(x) and ψ (i)(x), for i ∈ ℕ, denote the polygamma functions, where Γ(x) is the gamma function. In this paper, we prove that the functions
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$[\psi '(x)]^2 + \psi ''(x) - \frac{{x^2 + 12}} {{12x^4 (x + 1)^2 }}$$ \end{document}
and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{x + 12}} {{12x^4 (x + 1)}} - \{ [\psi '(x)]^2 + \psi ''(x)\}$$ \end{document}
are completely monotonic on (0,∞).
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