On the convexity of lemniscates

Let ${L_1}$ denote the lemniscate $|\prod \nolimits _{v = 1}^n {(z - {\zeta _v})| = 1}$. Assume the poles ${\zeta _v}$ are inscribed in the disc $|z| \leqq a$. Let ${z_0} = {n^{ - 1}}\sum \nolimits _{v = 1{\zeta _v}}^n {}$. Conditions for the convexity of ${L_1}$ are established in terms of $a$ and ${z_0}$. Sharp bounds are derived for real ${\zeta _v}$.