Let \(u\) be a positive measurable function on \([0,1]\). Which of the following is larger:

\[\int_{(0,1)} u(x) \log u(x) \lambda(d x) \quad \text { or } \quad \int_{(0,1)} u(s) \lambda(d s) \cdot \int_{(0,1)} \log u(t) \lambda(d t) ?\]

[ show that \(\log x \leqslant x \log x, x>0\), and assume first that \(\int u d \lambda=1\), then consider \(\left.u / \int u d \lambda.ight]\)