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## You are looking at 1 - 5 of 5 items for :

• "26D07"
Clear All  # On rational approximation to | x |

Author: Horst Alzer

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} $$e_n (S;x) = |x| - x\frac{{p(x) - p( - x)}} {{p(x) + p( - x)}},$$ \end{document}
We prove: for all n ≧ 1 and x ∈ [−1, 1] we have
\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} $$|e_n (S;x)| \leqq \frac{1}{{n^2 }},$$ \end{document}
where equality holds if and only if n = x = 1 or n = 1, x = −1. This refines a result of Brutman (1998), who showed 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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$|e_n (S;x)| < \frac{8} {{e^2 (n^2 - 1)}}$$ \end{document}
is valid for all n ≧ 2 and x ∈ [−1, 1].

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# Pasting Muckenhoupt weights through a contact point between sets of different dimensions

Authors: Hugo Aimar, Bibiana Iaffei and Liliana Nitti

## Abstract

We extend the result in  to the spaces of homogeneous type built in  starting from two sets of different dimensions with a unique point of contact.

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# On some functional inequalities related to the logarithmic mean

Author: W. Fechner

## Abstract

We deal with some functional inequalities which are motivated by the following relation between the geometric mean G, the logarithmic mean L and the arithmetic mean A:

\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} $$G^{\frac{2} {3}} \cdot A^{\frac{1} {3}} \leqq L \leqq \frac{2} {3}G + \frac{1} {3}A.$$ \end{document}

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

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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# The perimeter of rounded convex planar sets

Author: Csirmaz László

## Abstract

A convex set is inscribed into a rectangle with sides a and 1/a so that the convex set has points on all four sides of the rectangle. By “rounding” we mean the composition of two orthogonal linear transformations parallel to the edges of the rectangle, which makes a unit square out of the rectangle. The transformation is also applied to the convex set, which now has the same area, and is inscribed into a square. One would expect this transformation to decrease the perimeter of the convex set as well. Interestingly, this is not always the case. For each a we determine the largest and smallest possible increase of the perimeter.

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