Research article

The norm of pre-Schwarzian derivative on subclasses of bi-univalent functions

  • Received: 02 October 2018 Accepted: 28 November 2018 Published: 03 December 2018
  • MSC : 30C45

  • In the present paper, we give the best estimates for the norm of pre-Schwarzian derivatives $||{T_f}(z)|| = \mathop {\sup }\limits_{|z| < 1} (1 - |z{|^2})\left| {\frac{{f''(z)}}{{f'(z)}}} \right|$ for subclasses of bi-univalent functions.

    Citation: Shalini Rana, Pranay Goswami, Ravi Shanker Dubey. The norm of pre-Schwarzian derivative on subclasses of bi-univalent functions[J]. AIMS Mathematics, 2018, 3(4): 600-607. doi: 10.3934/Math.2018.4.600

    Related Papers:

  • In the present paper, we give the best estimates for the norm of pre-Schwarzian derivatives $||{T_f}(z)|| = \mathop {\sup }\limits_{|z| < 1} (1 - |z{|^2})\left| {\frac{{f''(z)}}{{f'(z)}}} \right|$ for subclasses of bi-univalent functions.


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    [1] S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J., 28 (1999), 217-230.
    [2] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babesş-Bolyai Math., 31 (1986), 70-77.
    [3] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, 1983.
    [4] H. Rahmatan, Sh. Najafzadeh and A. Ebadian, The norm of pre-Schwarzian derivatives on biunivalent functions of order α, B. Iran. Math. Soc., 43 (2017), 1037-1043.
    [5] T. S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.
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  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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