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• Author or Editor: Horst Alzer
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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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Представление в виде рядов γ и других математических констант

Authors: Horst Alzer and Stamatis Koumandos

Abstract

We present series representations for some mathematical constants, like γ, π, log 2, ζ(3). In particular, we prove that the following representation for Euler’s constant is valid:

\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} $$\gamma = \sum\limits_{r = 1}^\infty {\sum\limits_{s = 1}^r {\left( {\begin{array}{*{20}c} {r - 1} \\ {s - 1} \\ \end{array} } \right)( - 1)^{r - s} 2^s \left( {\frac{1} {s} + \log \frac{s} {{s + 1}}} \right)} } .$$ \end{document}

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Companions of the inequalities of Fejér--Jackson and Young

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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On the trigonometric polynomials of Fejér and Young

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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Series and product representations for some mathematical constants

Authors: Horst Alzer and Stamatis Koumandos

Abstract

We present several series and product representations for γ, π, and other mathematical constants. One of our results states that, for all real numbers µ s>0, 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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma = \sum\limits_{k = 0}^\infty {\frac{1} {{(1 + \mu )^{k + 1} }}\sum\limits_{m = 0}^k {\left( {_m^k } \right)} \left( { - 1} \right)^m \mu ^{k - m} S(m),}$$ \end{document}
where S(m) = ∑k=1 1/2k+m.

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Inequalities involving gamma and digamma functions

Authors: Horst Alzer and Chao-Ping Chen

We prove:

1. (A) 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} $$\Delta _c (x) = \log \frac{{\Gamma (x + 1)}}{{\sqrt {2\pi } (x/e)^x }} - \frac{1}{2}\psi (x + c) (x > 0; c \geqq 0).$$ \end{document}
1. (i) −Δc is completely monotonic on (0, ∞) if and only if c ≧ 2/3.
2. (ii) Δc is completely monotonic on (0, ∞) if and only if c = 0.
2. (B) The inequalities
\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} $$\frac{1}{2}\psi (x + a_0 ) < \log \frac{{\Gamma (x + 1)}}{{\sqrt {2\pi } (x/e)^x }} < \frac{1}{2}\psi (x + b_0 )$$ \end{document}
hold for all x > 0 with the best possible constants a 0 = 0.52660… and b 0 = 2/3.

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A variant of the Fejér–Jackson inequality

Authors: Horst Alzer and Man Kam Kwong

Abstract

We prove: For all natural numbers n and real numbers x ∈ [0, π] we have $−548130−585≤∑k=1n(−1)k+1(sin((2k−1)x)2k−1+sin(2kx)2k)$.

The sign of equality holds if and only if n = 2 and x = 4π/5.

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