Geodesic
A Logarithmic Inequality
Matematičeskie zametki, Tome 103 (2018) no. 2, pp. 210-222

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The inequality \begin{equation*} \ln\ln(r-\ln r)+1 \min_{0\le r-1} (\ln x+ x^{-1}\ln(r-x)) \ln\ln(r-\ln(r-2^{-1}\ln r))+1, \end{equation*} where $r>2$, is proved. A combinatorial optimization problem which involves the function to be minimized is described.
Keywords: logarithmic inequality, two-sided estimate, extremal graph. 
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     author = {G. V. Kalachev and S. Yu. Sadov},
     title = {A {Logarithmic} {Inequality}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {210--222},
     publisher = {mathdoc},
     volume = {103},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_103_2_a4/}
}
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G. V. Kalachev; S. Yu. Sadov. A Logarithmic Inequality. Matematičeskie zametki, Tome 103 (2018) no. 2, pp. 210-222. http://geodesic.mathdoc.fr/item/MZM_2018_103_2_a4/