Kodai Mathematical Journal

A class of univalent functions defined by a differential inequality

Milutin Obradović and Saminathan Ponnusamy

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Abstract

Let $\mathcal{A}$ be the class of analytic functions in the unit disk D with the normalization f(0) = f′(0) − 1 = 0. For λ > 0, denote by $\mathcal{M}$(λ) the class of functions f $\in$ $\mathcal{A}$ which satisfy the condition

$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq \lambda,\quad z\in \mathbf{D}.$

We show that functions in $\mathcal{M}$(1) are univalent in D and we present one parameter family of functions in $\mathcal{M}$(1) that are also starlike in D. In addition to certain inclusion results, we also present characterization formula, necessary and sufficient coefficient conditions for functions in $\mathcal{M}$(λ), and a radius property of $\mathcal{M}$(1).

Article information

Source
Kodai Math. J., Volume 34, Number 2 (2011), 169-178.

Dates
First available in Project Euclid: 5 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1309829544

Digital Object Identifier
doi:10.2996/kmj/1309829544

Mathematical Reviews number (MathSciNet)
MR2811638

Zentralblatt MATH identifier
1230.30010

Citation

Obradović, Milutin; Ponnusamy, Saminathan. A class of univalent functions defined by a differential inequality. Kodai Math. J. 34 (2011), no. 2, 169--178. doi:10.2996/kmj/1309829544. https://projecteuclid.org/euclid.kmj/1309829544


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