A Logarithmic Inequality

The inequality
\begin{equation*} \ln\ln(r-\ln r)+1 <\min_{0<x\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.