A Logarithmic Inequality

Abstract

The inequality

\(\ln {\kern 1pt} \ln \left( {r - \ln r} \right) + 1 < \mathop {\min }\limits_{0 < x \leqslant r - 1} \left( {\ln x + {x^{ - 1}}\ln \left( {r - x} \right)} \right) < \ln {\kern 1pt} \ln \left( {r - \ln \left( {r - {2^{ - 1}}\ln r} \right)} \right) + 1,\)

where r > 2, is proved. A combinatorial optimization problem which involves the function to be minimized is described.