Zalcman Conjecture for certain analytic and univalent functions

2020/06/14 by Allu, Vasudevarao, Pandey, Abhishek · 1 citation
#30C45 #30C50 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.07783

Abstract

Let A denote the class of analytic functions in the unit disk \mathbbD of the form f(z)= z+∑_n=2^∞a_n zⁿ and S denote the class of functions f\inA which are univalent (\it i.e., one-to-one). In 1960s, L. Zalcman conjectured that |a_n²-a_2n-1|≤ (n-1)² for n≥ 2, which implies the famous Bieberbach conjecture |a_n|≤ n for n≥ 2. For f∈ S, Ma \citeMa-1999 proposed a generalized Zalcman conjecture |a_na_m-a_n+m-1|≤ (n-1)(m-1) for n≥ 2, m≥ 2. Let U be the class of functions f\inA satisfying |f'(z)((z)/(f(z)))²-1 |lt; 1 for z∈\mathbbD. and F denote the class of functions f∈ A satisfying \rm Re (1-z)²f'(z)>0 in \mathbbD. In the present paper, we prove the Zalcman conjecture and generalized Zalcman conjecture for the class U using extreme point theory. We also prove the Zalcman conjecture and generalized Zalcman conjecture for the class F for the initial coefficients.

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