Show that the following inequality hold for all k≥1, real numbers a1,a2,...,ak, and positive numbers x1,x2,...,xk.
\ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} .
L. Losonczi inequalitieslogarithmsreal analysisreal analysis unsolved