A product theorem for $\mathcal {F}_p$ classes and an application

Abstract:

For Re $p > 0$ let ${\mathcal {F}_p} = \{ f|f(z) = \int _{|x| = 1} {{{(1 - xz)}^{ - p}}d\mu (x)}$, $|z| < 1$, $\mu$ a probability measure on $|x| = 1$ and let ${\mathcal {F}_p} \cdot {\mathcal {F}_q} = \{ fg|f \in {\mathcal {F}_p},g \in {\mathcal {F}_q}\}$. Brickman, Hallenbeck, MacGregor and Wilken proved a product theorem for the ${\mathcal {F}_p}$ classes; they showed that if $p > 0$, $q > 0$, then ${\mathcal {F}_p} \cdot {\mathcal {F}_q} \subset {\mathcal {F}_{p + q}}$. We give an (essentially complete) converse for the result of Brickman et al., i.e., we show that if ${\mathcal {F}_p}\cdot {\mathcal {F}_q} \subset {\mathcal {F}_{p + q}}$, then $p > 0$, $q > 0$ or else $p = q = 1 + it$ for some $t$ real. As an immediate consequence we disprove a conjecture about the extreme points of the closed convex hulls of the classes ${\text {Sp(}}\gamma {\text {)}}$, $0 < |\gamma | < \pi /2$, of $\gamma$-spirallike univalent functions, i.e., writing $m = 1 + {e^{ - 2i\gamma }}$, we show $\{ z/(1 - xz)^m| |x| = 1\} \subsetneqq \mathcal {E}\mathcal {K} \operatorname {Sp}(\gamma )$, $0 < |\gamma | < \pi /2$.