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

Summary: Let $𝒰\left(\lambda ,\mu \right)$ denote the class of all normalized analytic functions $f$ in the unit disk $|z|<1$ satisfying the condition

$\frac{f\left(z\right)}{z}\ne 0\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\left|{f}^{\text{'}}\left(z\right){\left(\frac{z}{f\left(z\right)}\right)}^{\mu +1}-1\right|<\lambda ,\phantom{\rule{2.em}{0ex}}\left|z\right|<1·$

For $f\in 𝒰\left(\lambda ,\mu \right)$ with $\mu \le 1$ and $0\ne {\mu }_{1}\le 1$, and for a positive real-valued integrable function $\varphi$ defined on $\left[0,1\right]$ satisfying the normalizing condition ${\int }_{0}^{1}\varphi \left(t\right)dt=1$, we consider the transform ${G}_{\varphi }f\left(z\right)$ defined by

${G}_{\varphi }f\left(z\right)=z{\left[{\int }_{0}^{1}\varphi \left(t\right){\left(\frac{zt}{f\left(tz\right)}\right)}^{\mu }dt\right]}^{-1/{\mu }_{1}},\phantom{\rule{2.em}{0ex}}z\in {\Delta }·$

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 $𝒰\left(\lambda ,\mu \right)$.

##### MSC:
 30C45 Special classes of univalent and multivalent functions
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