Norm of the Ces\`aro operator between some spaces of analytic functions

In this paper, we determine the exact norm of the Ces\`aro operator $\mathcal{C}$ on the Korenblum space $H^\infty_\alpha$ for $0 < \alpha \leq \frac12$ and on the logarithmically weighted space $H^\infty_{\alpha,\log}$ for $0 < \alpha < 1$. Moreover, we compute its norm when acting from $H^\infty_{\alpha,\log}$ to $H^\infty_\alpha$. Finally, we establish lower and upper bounds for the norm of $\mathcal{C}$ on the $\alpha$-Bloch space $\mathcal{B}^\alpha$ for $\alpha > 1$, and from the Hardy space $H^\infty$ to $\mathcal{B}^\alpha$ for $\alpha\geq 1$.