On a multiplier conjecture for univalent functions

Abstract:

Let $\mathcal {S}$ be the set of normalized univalent functions, and let $\mathcal {D}$ be the subset of $\mathcal {S}$ containing functions with the property: \[ \left | {f"(z)} \right | \leq {\text {Re}}f’(z),\quad \left | z \right | < 1\]. We present and discuss the following conjecture: For $f \in \mathcal {D},{\text {g,}}h \in \overline {{\text {co}}} (\mathcal {S})$, \[ {\text {Re}}\frac {1}{z}(f * {\text {g}} * h)(z) > 0,\quad \left | z \right | < 1\]. In particular, we prove that the conjecture holds with $\mathcal {S}$ replaced by $\mathcal {C}$, the class of close-to-convex functions, and show its truth for a number of special members of $\mathcal {D}$. These latter results are extensions of old ones of Szegà and Kobori about sections of univalent functions.