## 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.)

### Keywords:

Bazilevič function; starlike function; spirallike function
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### References:

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