Abstract
Lawrence Zalcman’s conjecture states that if \({f(z)=z+\sum\nolimits_{n=2}^{\infty}a_{n}z^{n}}\) is analytic and univalent in the unit disk \({|z|<1}\), then \({|a_n^2-a_{2n-1}|\leq (n-1)^2,}\) for each \({n\geq 2}\), with equality only for the Koebe function \({k(z)=z/(1-z)^2}\) and its rotations. This conjecture remains open although it has been verified for a few geometric subclasses of the class of univalent analytic functions. In this paper, we consider this problem for the family of normalized functions f analytic and univalent in the unit disk |z| < 1 satisfying the condition
Functions satisfying this condition are known to be convex in some direction (and hence close-to-convex and univalent) in |z| < 1. A few other related basic results and remarks about the Hayman index of functions in this family are also presented.
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S. Ponnusamy is on leave from the Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India.
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Muhanna, Y.A., Li, L. & Ponnusamy, S. Extremal problems on the class of convex functions of order −1/2. Arch. Math. 103, 461–471 (2014). https://doi.org/10.1007/s00013-014-0705-6
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DOI: https://doi.org/10.1007/s00013-014-0705-6