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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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Authors: Muhammad Aslam Noor, Gabriela Cristescu and Muhammad Uzair Awan

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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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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