On the coefficients of functions with bounded boundary rotation

Abstract:

Let ${V_k}$ be the class of normalised functions of bounded boundary rotation. For $f \in {V_k}$ define \[ M(r,f) = \max \limits _{|z| = r} |f(z)|,\] and let $L(r,f)$ denote the length of $f(|z| = r)$. Then if $f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}$, it is shown that (i) $2M(r,f) < L(r,f) \leqq k\pi M(r,f)$, and (ii) ${n^2}|{a_n}| \leqq (3k/{r^{n - 1}})M(r,f’),n \geqq 2$. The class ${\Lambda _k}$ of meromorphic functions of boundary rotation is also studied and estimates for the coefficients are given.