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The inequality $$\ln {\kern 1pt} \ln \left( {r - \ln r} \right) + 1 < \mathop {\min }\limits_{0 < x \leqslant r - 1} \left( {\ln x + {x^{ - 1}}\ln \left( {r - x} \right)} \right) < \ln {\kern 1pt} \ln \left( {r - \ln \left( {r - {2^{ - 1}}\ln r} \right)} \right) + 1,$$ ln ln ( r − ln r ) + 1 < min 0 < x ≤ r − 1 ( ln x + x − 1 ln ( r − x ) ) < ln ln ( r − ln ( r − 2 − 1 ln r ) ) + 1 , where r > 2, is proved. A combinatorial optimization problem which involves the function to be minimized is described.
Mathematical Notes – Springer Journals
Published: Mar 14, 2018
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