Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inequalities involving the gamma and digamma functions   We improve the upper bounds of the following inequalities proved in [H. Alzer
and N. Batir, Monotonicity properties of the gamma function, Appl. Math.
Letters, 20(2007), 778-781]. \begin{equation*}
exp\left(-\frac{1}{2}\psi\left(x+\frac{1}{3}\right)\right)<\frac{\Gamma(x)}{\sqrt{2\pi}x^xe^{-x}}<exp\left(-\frac{1}{2}\psi(x)\right),
\end{equation*} and $$ \frac{1}{2}\psi'(x+1/3))<\log
x-\psi(x)<\frac{1}{2}\psi'(x).$$ Here $\Gamma$ is the classical gamma function
and $\psi$ is the digamma function.

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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: stat
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The categories of this paper are: hep-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph