It is a question in one problem book:
Prove $\ln\frac{p}{q}\leq \frac{p-q}{\sqrt{pq}}$ for $0
.
Actually I already solved it: Define $F(x)=\frac{x-q}{\sqrt{xq}}-\ln x+\ln q$, then F'(x)\geq0 when $x\geq q$.
However,the problem book gives a hint to use Schwarz inequality $\left(\int_a^b f(x)g(x)dx\right)^2\leq \int_a^b f^{2}(x)dx\cdot\int_a^bg^2(x)dx$ I don't know how to use it.