  We are interested in the optimal growth in terms of $L^p$-averages of
hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some
weighted Taylor shift operators acting on the space of analytic function on the
unit disc. We unify the results obtained by considering intermediate notions of
upper frequent hypercyclicity between the $\mathcal{U}$-frequent hypercyclicity
and the hypercyclicity.
