  We study bounds for the backward shift operator $f \mapsto (f(z)-f(0))/z$ 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 $\|u\|_{h^1}$, where
$M_1$ is the integral mean with $p=1$.
