On the radius of convexity of linear combinations of univalent functions and their derivatives

✍ Scribed by Richard Greiner; Oliver Roth

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 169 KB
• Volume: 254-255
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310060

✦ Synopsis

Abstract

Let S denote the set of normalized univalent functions in the unit disk. We consider the problem of finding the radius of convexity r~α~ of the set

{(1 – α)f(z) + αzf′(z) : f ∈ S}

for fixed α ∈ ℂ. Using a linearization method we find the exact value of r~α~ for α ∈ [0, 1] and prove the (sharp) estimate r~α~ ≥ r~1~ for α ∈ ℂ with |2__α__– 1| ≤ 1. As an application of these results the sharp lower bound for the radius of convexity of the convolution f ∗︁ g where f, g ∈ S and g is close–to–convex is found to be 5 – 2√6. The case α = 1/2 is related to an old conjecture of Robinson dating back to 1947.