On the convexity of lemniscates

Abstract

Let L 1 {L_1} denote the lemniscate | ∏ v = 1 n ( z − ζ v ) | = 1 |\prod \nolimits _{v = 1}^n {(z - {\zeta _v})| = 1} . Assume the poles ζ v {\zeta _v} are inscribed in the disc | z | ≦ a |z| \leqq a . Let z 0 = n − 1 ∑ v = 1 ζ v n {z_0} = {n^{ - 1}}\sum \nolimits _{v = 1{\zeta _v}}^n {} . Conditions for the convexity of L 1 {L_1} are established in terms of a a and z 0 {z_0} . Sharp bounds are derived for real ζ v {\zeta _v} .