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Faber polynomial coefficients for certain subclasses of analytic and biunivalent functions. (English) Zbl 1444.30012
Summary: In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions defined in the open unit disc. We use the Faber polynomial expansions to find upper bounds for the \(n\)th (\(n\geq 3\)) Taylor-Maclaurin coefficients \(\left\vert a_n\right\vert\) of functions belong to these new subclasses with \(a_k=0\) for \(2\leq k\leq n-1\), also we find non-sharp estimates on the first two coefficients \(\left\vert a_2\right\vert\) and \(\left\vert a_3\right\vert \). The results, which are presented in this paper, would generalize those in related earlier works of several authors.
MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C55 General theory of univalent and multivalent functions of one complex variable
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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