Estimate for Schwarzian derivative of certain close-to-convex functions

Zhenyong Hu; Xiaoyuan Wang; Jinhua Fan; Zhenyong Hu; Xiaoyuan Wang; Jinhua Fan

doi:10.3934/math.2021626

AIMS Mathematics

2021, Volume 6, Issue 10: 10778-10788. doi: 10.3934/math.2021626

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Research article

Estimate for Schwarzian derivative of certain close-to-convex functions

1.
Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
2.
Institute of Sponge City Research, Pingxiang University, Pingxiang 337055, China

Received: 29 May 2021 Accepted: 12 July 2021 Published: 26 July 2021
MSC : 30C45

Let $ f(z) $ be analytic in the unit disk with $ f(0) = f'(0)-1 = 0 $. For the following close-to-convex subclasses: $ \Re \{(1-z)f'(z)\} > 0, $ $ \Re \{(1-z^{2})f'(z)\} > 0, $ $ \Re \{(1-z+z^{2})f'(z)\} > 0 $ and $ \Re \{(1-z)^{2}f'(z)\} > 0 $, we investigate the bounds for the first two consecutive derivatives of higher order Schwarzian derivatives of $ f(z) $.
- analytic functions,
- univalent functions,
- close-to-convex,
- higher order Schwarzian derivative
Citation: Zhenyong Hu, Xiaoyuan Wang, Jinhua Fan. Estimate for Schwarzian derivative of certain close-to-convex functions[J]. AIMS Mathematics, 2021, 6(10): 10778-10788. doi: 10.3934/math.2021626

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Abstract

Let $ f(z) $ be analytic in the unit disk with $ f(0) = f'(0)-1 = 0 $. For the following close-to-convex subclasses: $ \Re \{(1-z)f'(z)\} > 0, $ $ \Re \{(1-z^{2})f'(z)\} > 0, $ $ \Re \{(1-z+z^{2})f'(z)\} > 0 $ and $ \Re \{(1-z)^{2}f'(z)\} > 0 $, we investigate the bounds for the first two consecutive derivatives of higher order Schwarzian derivatives of $ f(z) $.

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