For $a_1,...,a_n,b_1,...,b_n>0,\quad$ define $a:=\sum a_i,\ b:=\sum b_i,\ s:=\sum \sqrt{a_ib_i}$.
Is the following inequality true?: $${\frac{\Bigl(\prod a_i^{a_i}\Bigr)^\frac1a}a \cdot \frac{\left(\prod b_i^{b_i}\right)^\frac1b}b \le\left(\frac{\prod (a_ib_i)^{ \sqrt{a_ib_i}}}{s^{2s}}\right)^\frac s{ab}}.$$ If true, how can I prove it? And if not true, is there a counter-example?
An alternate formulation of this inequality, as suggested by cardinal is
$$\sum_i x_i^2 \log x_i^2 + \sum_i y_i^2 \log y_i^2 \leq \rho^2 \sum_i \frac{x_iy_i}{\rho} \log \frac{x_iy_i}{\rho} \; ,$$
in which $x_i,y_i>0$, $\sum_i x_i^2 = \sum_i y_i^2 = 1$ and $\sum_i x_i y_i = \rho$. It can be found by taking the logarithm of the original expression and defining $x_i=\sqrt{a_i/a}$ and $y_i=\sqrt{b_i/b}$.