Let \(f\) be holomorphic in a neighborhood of \(D_r(z_0)\). Show that for any \(s < r\), there exists a constant \(c > 0\) such that \[ \norm{f}_{(\infty,s)} \le c\norm{f}_{(1,r)}, \] where \(\norm{f}_{(\infty,s)} = \sup_{z\in D_s(z_0)} \abs{f(z)}\) and \(\norm{f}_{(1,r)} = \int_{D_r(z_0)} \abs{f(z)}\,dx\,dy\).
The source adds “Note: Exercise 3.8.20 on p.107 in Stein et al is a straightforward consequence of this stronger result using the integral form of the Cauchy-Schwarz inequality in real analysis.”, a pointer to Stein and Shakarchi, Complex Analysis; the statement does not depend on it.