Hull subordination and extremal problems for starlike and spirallike mappings

Abstract:

Let $\mathfrak {F}$ be a compact subset of the family $\mathcal {A}$ of functions analytic in $\Delta = \{ z:\;|z| < 1\}$, and let $\mathcal {L}$ be a continuous linear operator of order zero on $\mathcal {A}$. We show that if the extreme points of the closed convex hull of $\mathcal {F}$ is the set $\{ {f_0}(xz)\} (|x| = 1)$, then $\mathcal {L}(f)$ is hull subordinate to $\mathcal {L}({f_0})$ in $\Delta$. This generalizes results of R. M. Robinson corresponding to families $\mathcal {F}$ of functions that are subordinate to $(1 + z)/(1 - z)$ or to $1/{(1 - z)^2}$. Families $\mathcal {F}$ to which this theorem applies are discussed and we identify each such operator $\mathcal {L}$ with a suitable sequence of complex numbers. Suppose that $\Phi$ is a nonconstant entire function and that $0 < |{z_0}| < 1$. We show that the maximum of $\operatorname {Re} \{ \Phi [\log (f({z_0})/{z_0})]\}$ over the class of starlike functions of order a is attained only by the functions $f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;|x| = 1$. A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.