Let ${\cal U}=\{z\in\mathbb{C} :|z|<1\}$ be the unit disc in the complex plane.Let $\Sigma_k$ be the class of meromorphic functions $f$ in $\cal U$ having the form:$$f(z)=\frac{1}{z}+\alpha_kz^k+\cdots,0<|z|<1,k\geq0$$ A function $f\in\Sigma=\Sigma_0$ is called starlike if $$\Re\left[-\frac{zf'(z)}{f(z)}\right]>0\;\mbox{in}\;\cal U$$ Let denote by $\Sigma_k^*$ the class of starlike functions in$\Sigma_k$ and by $A_n$ the class of holomorphic functions $g$ of the form: $$g(z)=z+a_{n+1}z^{n+1}+\cdots\;,z\in{\cal U},n\geq1$$ With suitable conditions on $k,p\in\mathbb{N}$,on $c\in\mathbb{R}$,on $\gamma\in\mathbb{C}$ and on the function $g\in A_{k+1}$, the author shows that the integral operator\\ $L_{g,c,\gamma}:\Sigma\to\Sigma$ defined by: $$K_{g,c}(f)(z)\equiv \frac{c}{g^{c+1}(z)}\int_0^zf(t)g^c(t)\mathrm{e}^{\gamma t^p}dt,z\in{\cal U},f\in\Sigma$$ maps $\Sigma_k^*$ into $\Sigma_l^*$, where $l=\mathrm{min}\{p-1,k\}.$

### A meromorphic-starlikeness-preserving property of an integral operator

#### Abstract

Let ${\cal U}=\{z\in\mathbb{C} :|z|<1\}$ be the unit disc in the complex plane.Let $\Sigma_k$ be the class of meromorphic functions $f$ in $\cal U$ having the form:$$f(z)=\frac{1}{z}+\alpha_kz^k+\cdots,0<|z|<1,k\geq0$$ A function $f\in\Sigma=\Sigma_0$ is called starlike if $$\Re\left[-\frac{zf'(z)}{f(z)}\right]>0\;\mbox{in}\;\cal U$$ Let denote by $\Sigma_k^*$ the class of starlike functions in$\Sigma_k$ and by $A_n$ the class of holomorphic functions $g$ of the form: $$g(z)=z+a_{n+1}z^{n+1}+\cdots\;,z\in{\cal U},n\geq1$$ With suitable conditions on $k,p\in\mathbb{N}$,on $c\in\mathbb{R}$,on $\gamma\in\mathbb{C}$ and on the function $g\in A_{k+1}$, the author shows that the integral operator\\ $L_{g,c,\gamma}:\Sigma\to\Sigma$ defined by: $$K_{g,c}(f)(z)\equiv \frac{c}{g^{c+1}(z)}\int_0^zf(t)g^c(t)\mathrm{e}^{\gamma t^p}dt,z\in{\cal U},f\in\Sigma$$ maps $\Sigma_k^*$ into $\Sigma_l^*$, where $l=\mathrm{min}\{p-1,k\}.$
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/2553630
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