This argument is from [Eremenko-Lyubich survey][1]: Let $f$ be a rational map and $\mu$ an $f$-invariant ergodic measure on $\widehat{\Bbb{C}}$. By [Ledrappier-Young entropy formula][2], $h_\mu(f)$ is the product of the Lyapunov exponent $\chi_\mu$ by the dimension of the measure. Now pick an invariant ergodic measure with positive entropy supported in the Julia set; e.g. the measure of maximal entropy $\mu_f$ whose entropy is $\log(\deg f)$. We deduce that 
$$
\dim\mu_f=\frac{\log(\deg f)}{\chi_{\mu_f}}
$$  
is non-zero. Now notice that the Hausdorff dimension of $\mathcal{J}(f)={\rm{supp}}\mu_f$ cannot be smaller than the dimension of $\mu_f$.


  [1]: https://www.math.purdue.edu/~eremenko/dvi/AAsurveyp2.pdf
  [2]: http://www.scholarpedia.org/article/Pesin_entropy_formula#The_Ledrappier-Young_Entropy_Formula