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Differential Inequalities and Univalent Functions

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Abstract

Let \(\mathcal{M}\) be the class of analytic functions in the unit disk \(\mathbb{D}\) with the normalization f(0) = f′(0) − 1 = 0, and satisfying the condition

$$\left|{{z^2}{{\left({{z\over{f(z)}}}\right)}^{\prime\prime}}\;+\;f'(z){{\left({{z\over{f(z)}}} \right)}^2}\;-\;1}\right|\le 1,\;\;\;z\;\in\;\mathbb{D}.$$

Functions in \(\mathcal{M}\) are known to be univalent in \(\mathbb{D}\). In this paper, it is shown that the harmonic mean of two functions in \(\mathcal{M}\) are closed, that is, it belongs again to \(\mathcal{M}\). This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in \(\mathcal{M}\) are shown to be starlike in \(\mathbb{D}\). However we conjecture that functions in \(\mathcal{M}\) are not necessarily starlike, as apparently supported by other examples.

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Funding

R.M. Ali gratefully acknowledged support from a Universiti Sains Malaysia research university grant 1001/PMATHS/8011101. The work of the third author is supported by Mathematical Research Impact Centric Support of DST, India (MTR/2017/000367).

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Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, 11800, Malaysia

    Rosihan M. Ali

  2. Department of Mathematics, Faculty of Civil Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade, 11000, Serbia

    Milutin Obradović

  3. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    Saminathan Ponnusamy

Authors
  1. Rosihan M. Ali
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  2. Milutin Obradović
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  3. Saminathan Ponnusamy
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Corresponding authors

Correspondence to Rosihan M. Ali, Milutin Obradović or Saminathan Ponnusamy.

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Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Ali, R.M., Obradović, M. & Ponnusamy, S. Differential Inequalities and Univalent Functions. Lobachevskii J Math 40, 1242–1249 (2019). https://doi.org/10.1134/S1995080219090038

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  • DOI: https://doi.org/10.1134/S1995080219090038

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