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Univalence of integral operators on neighborhoods of analytic functions. (English) Zbl 1397.30013

Summary: Assume that \(\Delta\) is the unit disk in the complex plane and \(\mathcal{A}\) is the class of analytic functions \(f\) in \(\Delta\) with normalization conditions \(f(0)=f'(0)-1=0\). For \(\lambda_i,\mu_i\in \mathbb{C}\) and \(f_i\in\mathcal{A}(1\leq i\leq n)\), consider the integral operator: \[ \begin{aligned} F(z):=F_{\lambda,\mu}[(f_1,\ldots,f_n)](z)=\int_0^z \prod_{i=1}^n(f_i'(t))^{\lambda _{i}}\bigg(\frac{f_{i}(t)}{t}\bigg)^{\mu_{i}}\text{d}t \qquad (z\in\Delta),\end{aligned} \] where \(\lambda=(\lambda_1,\ldots,\lambda_n)\) and \(\mu=(\mu_1,\ldots,\mu_n)\). For \(\delta>0\), define \[ \begin{aligned} V_\delta(f):=\{g\in\mathcal{A}^n:\|f'-g'\|_\infty\leq\delta\}, \end{aligned} \] where \(\|f'-g'\|_\infty:=\max_{z\in\Delta,\,1\leq i\leq n}|f_i'(z)-g'_i(z)|\), a neighborhood of \(f\), \(f=(f_1,\ldots,f_n)\), \(g=(g_1,\ldots,g_n)\), \(f\in\mathcal{A}^n\), \(\mathcal{A}^n=\mathcal{A}\times\cdots\mathcal A\) and \(\times\) is the Cartesian product. In this paper, we determine the radii of \(V_\delta(f)\), such that the integral operator \(F(z)\) carries the neighborhood into the class \(\mathcal{S}\) (class of univalent functions), where \(f_i(1\leq i\leq n)\) belongs to the universal linear invariant families or satisfies certain conditions.

MSC:

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