Abstract
Let f be analytic in the unit disk D={z∈C:|z|<1}, and S be the subclass of normalized univalent functions with f(0)=0, and f′(0)=1. Let F be the inverse function of f, given by F(z)=ω+∑n=2∞Anωn for some |ω|≤r0(f). Let S*⊂S be the subset of starlike functions in D, and C the subset of convex functions in D. We show that −1≤|A3|−|A2|≤3 for f∈S, the upper bound being sharp, and sharp upper and lower bounds for |A3|−|A2| for the more important subclasses of S* and C, and for some related classes of Bazilevič functions.
Highlights
IntroductionIn 1985, de Branges [1] solved the famous Bieberbach conjecture by showing that if f ∈ S , | an | ≤ n for n ≥ 2, with equality for Koebe function f (z) = k (z) := z/(1 − z) or its rotation
Let A denote the class of analytic functions f in the unit disk D = {z ∈ C : |z| < 1} normalized by f (0) = 0 = f 0 (0) − 1
We show that −1 ≤ | A3 | − | A2 | ≤ 3 for f ∈ S, the upper bound being sharp, and sharp upper and lower bounds for | A3 | − | A2 | for the more important subclasses of S ∗ and C, and for some related classes of Bazilevič functions
Summary
In 1985, de Branges [1] solved the famous Bieberbach conjecture by showing that if f ∈ S , | an | ≤ n for n ≥ 2, with equality for Koebe function f (z) = k (z) := z/(1 − z) or its rotation It was, natural to ask: if for f ∈ S , the inequality || an+1 | − | an || ≤ 1 is true when n ≥ 2?. Apart from the inequalities for n = 2 above, there appears to be no known sharp upper or lower bounds for | an+1 | − | an | when n ≥ 3 for functions in S. n =2 valid on some disk |ω | ≤ r0 ( f ). In [9], the present authors gave sharp upper and lower bounds for | a3 | − | a2 |, when f belongs to the most important subclasses of starlike and convex functions in D.
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