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Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions. (English) Zbl 1182.30017
Summary: Let $\Cal U(\lambda , \mu )$ denote the class of all normalized analytic functions $f$ in the unit disk $|z| < 1$ satisfying the condition $$\frac{f(z)}{z} \ne 0 \quad \text{and} \quad \left|{f'(z)\left( {\frac{z}{f(z)}} \right)^{\mu + 1} - 1} \right| < \lambda ,\qquad \left| z \right| < 1.$$ For $f\in \Cal U(\lambda,\mu)$ with $\mu\le 1$ and $0\not=\mu_1\le 1$, 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 $\Cal U(\lambda,\mu)$.

30C45Special classes of univalent and multivalent functions
Full Text: DOI
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