>Let $\phi$ be a real smooth superharmonic function on unit disc $D$ in $\mathbb C$; i.e. $\triangle \phi\le 0$.
Then there is a curve $\gamma$ from the center of $D$ to its boundary such that 
$$\int\limits_\gamma e^\phi<\infty.$$

The question came from my failed answer [to this question][1].
I know that the answer is YES, but I do not see a direct proof.


  [1]: http://mathoverflow.net/questions/43614