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  • Author or Editor: Horst Alzer x
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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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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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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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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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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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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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Abstract

We prove: For all natural numbers n and real numbers x ∈ [0, π] we have 548130585k=1n(1)k+1(sin((2k1)x)2k1+sin(2kx)2k).

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

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