Bounds on the Norm of the Backward Shift and Related Operators in Hardy and Bergman Spaces

We study bounds for the backward shift operator $f\mapsto (f(z)-f(0))\mathrm{/}z$ and the related operator $f\mapsto f-f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound on $M_1(r,u-u(0))$ in terms of $\mathrm{\parallel }u{\mathrm{\parallel }}_{h^1}$, where $M_1$ is the integral mean with $p=1$.