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Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions. (English) Zbl 1182.30017

Summary: Let \(\mathcal U(\lambda , \mu )\) denote the class of all normalized analytic functions \(f\) in the unit disk \(|z| < 1\) satisfying the condition \[ \frac{f(z)}{z} \neq 0 \quad \text{and} \quad \left|{f'(z)\left( {\frac{z}{f(z)}} \right)^{\mu + 1} - 1} \right| < \lambda ,\qquad \left| z \right| < 1. \]
For \(f\in \mathcal U(\lambda,\mu)\) with \(\mu\leq 1\) and \(0\not=\mu_1\leq 1\), and for a positive real-valued integrable function \(\varphi\) defined on \([0,1]\) satisfying the normalizing condition \(\int_0^1\varphi(t)dt=1\), we consider the transform \(G_\varphi f(z)\) defined by
\[ G_\varphi f(z)=z\left[\int_0^1\varphi(t)\left(\frac{zt}{f(tz)}\right)^\mu dt\right]^{-1/\mu_1},\qquad z\in \Delta. \]
In this paper, we find conditions on the range of the parameters \(\lambda\) and \(\mu\) so that the transform \(G_\varphi f\) is univalent or starlike. In addition, for a given univalent function of a certain form, we provide a method for obtaining functions in the class \(\mathcal U(\lambda,\mu)\).

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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