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On the radius of convexity of linear combinations of univalent functions and their derivatives
Author(s) -
Greiner Richard,
Roth Oliver
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310060
Subject(s) - mathematics , convexity , conjecture , combinatorics , unit disk , radius , regular polygon , convolution (computer science) , mathematical analysis , unit (ring theory) , upper and lower bounds , pure mathematics , geometry , computer security , machine learning , computer science , artificial neural network , financial economics , economics , mathematics education
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.