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Authors: Kuei-Lin Tseng, Shiow-Ru Hwang and S. Dragomir

Abstract  

In this paper, we establish some new refinements for the celebrated Fejér’s and Hermite-Hadamard’s integral inequalities for convex functions.

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Abstract  

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus

\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} $$\mathbb{T}^n$$ \end{document}
:= [0, 2π)n, let
\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} $$\hat f$$ \end{document}
(k) denote the Fourier coefficient of f, where k = (k 1, … k n) ∈ ℤn. In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π]n to ℜ in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Móricz [5] and Fülöp and Móricz [3].

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Abstract  

By introducing some parameters, we give new extensions of Hilbert inequality with best constant factors.

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Abstract  

The quasilinearity of certain composite functionals associated to Schwarz’s celebrated inequality for inner products is investigated. Applications for operators in Hilbert spaces are given as well.

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Abstract  

The weighted averages of a sequence (c k), c k ∈ ℂ, with respect to the weights (p k), p k ≥ 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ℝ+ → ℂ with respect to the weight function p(t) ≥ 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit σ nL as n → ∞ or σ(t) → L as t → ∞ exists, respectively. These characterizations may find applications in probability theory.

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Abstract  

Jensen-Steffensen type inequalities for P-convex functions and functions with nondecreasing increments are presented. The obtained results are used to prove a generalization of Čebyšev’s inequality and several variants of H�lder’s inequality with weights satisfying the conditions as in the Jensen-Steffensen inequality. A few well-known inequalities for quasi-arithmetic means are generalized.

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Authors: V. Čuljak, B. Ivanković and J. Pečarić

Abstract  

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.

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Abstract  

We discuss and complement the knowledge about generalized Orlicz classes

\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} $$\tilde X_\Phi$$ \end{document}
and Orlicz spaces X Φ obtained by replacing the space L 1 in the classical construction by an arbitrary Banach function space X. Our main aim is to focus on the task to study inequalities in such spaces. We prove a number of new inequalities and also natural generalizations of some classical ones (e.g., Minkowski’s, H�lder’s and Young’s inequalities). Moreover, a number of other basic facts for further study of inequalities and function spaces are included.

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