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# Some sharp inequalities for n-monotone functions

Acta Mathematica Hungarica
Author: Petar P. Petrov
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# Multidimensional Ostrowski inequalities, revisited

Acta Mathematica Hungarica
Author: George Anastassiou
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# On the trigonometric polynomials of Fejér and Young

Periodica Mathematica Hungarica
Authors: Horst Alzer and Qinghe Yin

## Abstract

The trigonometric polynomials of Fejér and Young are defined by

\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} $$S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}}$$ \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} $$C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}$$ \end{document}
, respectively. We prove that the inequality
\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} $$\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}}$$ \end{document}
holds for all n ≥ 2 and x ∈ (0, π). The lower bound is sharp.

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# Companions of the inequalities of Fejér--Jackson and Young

Analysis Mathematica
Authors: Horst Alzer and Stamatis Koumandos

Summary Applications of  some well-known theorems  of Jackson and Young lead to the sharp inequalities -1<n k-1S(cos(kx)+sin(kx))/(n =1; 1<x<p)  and  -1/2Si(p)<n k-1S(cos(kx)·sin(kx))/(n =1; x?R)  We prove that the following counterpart is valid for all integers  n =1 and real numbers x? (0, p):  -3/2=n k-1S(cos(kx)-sin(kx))/k  where the sign of equality holds if and only if n =2 and x = p /2.

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# Approximations in several variables with freud-type and A p-weights

Studia Scientiarum Mathematicarum Hungarica
Author: Nguyen Ky

inequalities for weighted polynomial approximations, East J. Approx. , 12,3 (2006), 367–379. Ky N. X. Sharp inequalities for weighted polynomial approximations

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