Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The best bounds for Toader mean in terms of the centroidal and
  arithmetic means   In the paper, the authors discover the best constants $\alpha_{1}$,
$\alpha_{2}$, $\beta_{1}$, and $\beta_{2}$ for the double inequalities $$
\alpha_{1}\bar{C}(a,b)+(1-\alpha_{1}) A(a,b)< T(a,b) <\beta_{1}
\bar{C}(a,b)+(1-\beta_{1})A(a,b) $$ and $$
\frac{\alpha_{2}}{A(a,b)}+\frac{1-\alpha_{2}}{\bar{C}(a,b)}<\frac1{T(a,b)}
<\frac{\beta_{2}}{A(a,b)}+\frac{1-\beta_{2}}{\bar{C}(a,b)} $$ to be valid for
all $a,b>0$ with $a\ne b$, where $$
\bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)},\quad A(a,b)=\frac{a+b}2, $$ and
$$
T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}\,\td\theta
$$ are respectively the centroidal, arithmetic, and Toader means of two
positive numbers $a$ and $b$. As an application of the above inequalities, the
authors also find some new bounds for the complete elliptic integral of the
second kind.

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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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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: adap-org
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: physics