On a multiplier conjecture for univalent functions

Abstract

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