Characterization and the pre-Schwarzian norm estimate for concave univalent functions |
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Authors: | Bappaditya Bhowmik Saminathan Ponnusamy Karl-Joachim Wirths |
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Institution: | 1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India 2. Institut für Analysis, TU Braunschweig, 38106, Braunschweig, Germany
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Abstract: | Let Co(α) denote the class of concave univalent functions in the unit disk ${\mathbb{D}}$ . Each function ${f\in Co(\alpha)}$ maps the unit disk ${\mathbb{D}}$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional ${(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ . In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional ${(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H p space for p < 1/α. |
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