The growth rate of an entire function and the Hausdorff dimension of its   Julia set
  Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract. 
Walter Bergweiler, Bogus{\l}awa Karpi\'nska, Gwyneth M. Stallard