In this note we obtain by purely geometric means that for convex cocompact Kleinian groups the exponent of convergence is bounded from above by an expression which depends mainly on the diameter of the convex core of the associated infinite-volume hyperbolic manifold. This result is derived via refinements of Sullivan's shadow lemma and of estimates for the growth of the orbital counting function and Poincare series. We finally obtain spectral and fractal implications, such as lower bounds for the bottom of the spectrum of the Laplacian on these manifolds, and upper bounds for the decay of the area of neighbourhoods of the associated limit sets.