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The \(\mathcal {U}\)-Radius for Classes of Analytic Functions

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Abstract

Let \(\mathcal {U}\) denote the class of normalized analytic functions \(f\) in the open unit disk \(\mathbb {D}\) satisfying

$$\begin{aligned} \left| \left( \frac{z}{f(z)}\right) ^2f'(z)-1\right| < 1. \end{aligned}$$

The \(\mathcal {U}\)-radius is obtained for several classes of functions. These include the class of normalized analytic functions \(f\) satisfying the inequality \({{\mathrm{Re}}}\, f(z)/g(z)>0\) or \(\left| f(z)/g(z)-1\right| < 1\) in \(\mathbb {D},\) where \(g\) belongs to a certain class of functions, the class of functions \(f\) satisfying \(|f'(z)-1|<1\) in \(\mathbb {D},\) and functions \(f\) satisfying \({{\mathrm{Re}}}\, f(z)/z>\alpha , 0 \le \alpha < 1,\) in \(\mathbb {D}.\) A recent conjecture by Obradović and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true.

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Acknowledgments

This work benefited greatly from the stimulating discussions held with Prof. M. Obradović, Prof. S. Ponnusamy, and Prof. Y. Abu Muhanna. The work presented here was supported in parts by a research university grant 1001/PMATHS/811280 from Universiti Sains Malaysia.

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Authors and Affiliations

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

    Rosihan M. Ali

  2. Department of Mathematics, University of Dammam, Dammam, 31113, Kingdom of Saudi Arabia

    Najla M. Alarifi

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  1. Rosihan M. Ali
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  2. Najla M. Alarifi
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Correspondence to Rosihan M. Ali.

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Communicated by Lee See Keong.

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Ali, R.M., Alarifi, N.M. The \(\mathcal {U}\)-Radius for Classes of Analytic Functions. Bull. Malays. Math. Sci. Soc. 38, 1705–1721 (2015). https://doi.org/10.1007/s40840-015-0115-3

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  • DOI: https://doi.org/10.1007/s40840-015-0115-3

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