135 research outputs found
On logarithmic coefficients of some close-to-convex functions
The logarithmic coefficients of an analytic and univalent function
in the unit disk with the
normalization is defined by . Recently, D.K. Thomas [On the logarithmic
coefficients of close to convex functions, {\it Proc. Amer. Math. Soc.} {\bf
144} (2016), 1681--1687] proved that for functions
in a subclass of close-to-convex functions (with argument ) and claimed that
the estimate is sharp by providing a form of a extremal function. In the
present paper, we pointed out that such extremal functions do not exist and the
estimate is not sharp by providing a much more improved bound for the whole
class of close-to-convex functions (with argument ). We also determine a
sharp upper bound of for close-to-convex functions (with argument
) with respect to the Koebe function.Comment: 13 page
Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions
Let denote the class of analytic and univalent ({\it i.e.},
one-to-one) functions in the unit disk
. For , Ma proposed the
generalized Zalcman conjecture that |a_{n}a_{m}-a_{n+m-1}|\le
(n-1)(m-1),\,\,\,\mbox{ for } n\ge2,\, m\ge 2, with equality only for the
Koebe function and its rotations. In this paper using the
properties of holomorphic motion and Krushkal's Surgery Lemma
\cite{Krushkal-1995}, we prove the generalized Zalcman conjecture when ,
and , .Comment: 14 pages. arXiv admin note: text overlap with arXiv:2209.1059
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