Coefficient characterizations and sections for some univalent functions |
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Authors: | M Obradovi? S Ponnusamy K -J Wirths |
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Institution: | 1. University of Belgrade, Belgrade, Serbia 2. Indian Statistical Institute, Taramani, Chennai, India 3. Institute for Analysis and Algebra, TU Braunschweig, Braunschweig, Germany
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Abstract: | Let (α) denote the class of locally univalent normalized analytic functions f in the unit disk |z| < 1 satisfying the condition $Re\left( {1 + \frac{{zf''(z)}} {{f'(z)}}} \right) < 1 + \frac{\alpha } {2}for|z| < 1 $ and for some 0 < α ≤ 1. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients a n of f ∈ (α). Secondly, we determine the sharp bound for the Fekete-Szegö functional for functions in (α) with complex parameter λ. Thirdly, we present a convolution characterization for functions f belonging to (α) and as a consequence we obtain a number of sufficient coefficient conditions for f to belong to (α). Finally, we discuss the close-to-convexity and starlikeness of partial sums of f ∈ (α). In particular, each partial sum s n (z) of f ∈ (1) is starlike in the disk |z| ≤ 1/2 for n ≥ 11. Moreover, for f ∈ (1), we also have Re(s′ n (z)) > 0 in |z| ≤ 1/2 for n ≥ 11. |
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