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Sibirskii Matematicheskii Zhurnal, 2013, Volume 54, Number 4, Pages 852–870
(Mi smj2462)
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This article is cited in 40 scientific papers (total in 40 papers)
Coefficient characterizations and sections for some univalent functions
M. Obradovića, S. Ponnusamyb, K.-J. Wirthsc a Department of Mathematics, Faculty of Civil Engineering, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
b Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600113 India
c Institut für Analysis und Algebra, TU Braunschweig, Braunschweig 38106 Germany
Abstract:
Let $\mathscr G(\alpha)$ denote the class of locally univalent normalized analytic functions $f$ in the unit disk $|z|<1$ satisfying the condition
$$
\mathrm{Re}\left(1+\frac{zf''(z)}{f'(z)}\right)<1+\frac\alpha2\qquad\text{for}\quad|z|<1
$$
and for some $0<\alpha\le1$. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients $a_n$ of $f\in\mathscr G(\alpha)$. Secondly, we determine the sharp bound for the Fekete–Szegö functional for functions in $\mathscr G(\alpha)$ with complex parameter $\lambda$. Thirdly, we present a convolution characterization for functions $f$ belonging to $\mathscr G(\alpha)$ and as a consequence we obtain a number of sufficient coefficient conditions for $f$ to belong to $\mathscr G(\alpha)$. Finally, we discuss the close-to-convexity and starlikeness of partial sums of $f\in\mathscr G(\alpha)$. In particular, each partial sum $s_n(z)$ of $f\in\mathscr G(1)$ is starlike in the disk $|z|\le1/2$ for $n\ge11$. Moreover, for $f\in\mathscr G(1)$, we also have $\mathrm{Re}(s'_n(z))>0$ in $|z|\le1/2$ for $n\ge11$.
Keywords:
analytic function, univalent function, starlike function, close-to-convex function, convex function, coefficient inequality, area theorem, radius of univalency, subordination, convolution, Fekete–Szegö functional.
Received: 20.09.2012
Citation:
M. Obradović, S. Ponnusamy, K.-J. Wirths, “Coefficient characterizations and sections for some univalent functions”, Sibirsk. Mat. Zh., 54:4 (2013), 852–870; Siberian Math. J., 54:4 (2013), 679–696
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https://www.mathnet.ru/eng/smj2462 https://www.mathnet.ru/eng/smj/v54/i4/p852
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