We investigate the boundedness of the H$^\infty$-calculus by estimating the bound b(ε) of the mapping H$^\infty$→B(X): f→f(A)T(ε) for ε near zero. Here, −A generates the analytic semigroup T and H$^\infty$ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε) =O(|logε|) in general, whereas b(ε) =O(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(ε) =O(|logε|).