SCHWEITZER 1958 Problem 9

Show that if $f(z) = 1+a_1 z+a_2z^2+\dots$ for $\mid z \mid\leq 1$ and $\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2$ ,

then $f(z)$ has a root in the disc $\mid z \mid \leq 1$ .(F. 4)