For hyperbolic rational maps we show that the Hausdorff dimension of the associated Julia set is bounded away from 2, where the bounds depend exclusively on certain intrinsic geometric exponents. This result is derived via lower estimates for the iterate-counting-function and for the dynamical Poincare series. Subsequently, we deduce some interesting consequences, such as upper bounds for the decay of the area of parallel-neighbourhoods of the Julia set, and lower bounds for the Lyapunov exponents with respect to the measure of maximal entropy.