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• 1 ]Morsbacher Str. 10 51545 Waldbröl Germany
• | 2 Henan Polytechnic University School of Mathematics and Informatics Jiaozuo City 454003 Henan Province, People’s Republic of China
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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 a0 = 0.52660… and b0 = 2/3.

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Studia Scientiarum Mathematicarum Hungarica
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