I am stuck with an exercise on the relation between KL and Kakutani-Hellinger distance, i.e. the objective is to show $$K(P,\tilde{P}) \geq -2\log(1-\rho^2(P,\tilde{P}))$$ wherein $K$ denotes the KL-"distance" defined via $$E_P(\log(z))$$ (z denoting the Radon-Nikodym derivative of $dP$ with respect to $d\tilde{P}$assuming $dP << d\tilde{P}$) and wherein the term $(1-\rho^2(P,\tilde{P}))$ denotes the Hellinger integral $$E_{\tilde{P}}\left(\sqrt{z}\right).$$
This boils down to showing the following: $$E_{\tilde{P}}\left(z \log(z)\right) \geq -2\log E_{\tilde{P}}\left(\sqrt{z}\right)$$
I need to employ an appropriate inequality between these functions - but I have yet no idea on how to specify an appropriate relation.