Meromorpc starlikeness preserved by a general class of integral operators

Abstract

Let $\mathbb D$ be the complex unit disc and let $\Sigma_k$ be the class of all meromorphic functions $f\in\mathbb D\setminus\{0\}$ of the form:
$$f(z)=\frac{1}{z}+a_kz^k+a_{k+1}z^{k+1}+\cdots\;k\in\mathbb N,\;a_k\neq 0.$$
A function in $\Sigma:=\Sigma_0$ is called (meromorphic) starlike if
$$\mathrm{Re}\left[ -\frac{zf^{\prime }(z)}{f(z)}\right] >0 \text{ in}\;\mathbb D\setminus\{0\}$$
and $\Sigma_k^*$ is the subclass of starlike (meromorphic) functions in $\Sigma_k$. The purpose of the article is to improve
many previous results by giving suffcient conditions on the (given) analytic functions (in $\mathbb D$) $g$ and $h$ and on
the numbers $k,m\in\mathbb N$ and $c>0$ so that the integral operator
$$I_{g,h}^c(f)(z)=\frac{c}{g^{c+1}(z)}\int_0^zf(t)g^c(t)h(t)dt,\;z\in\mathbb D$$
is well-defined in $\Sigma$ and maps $\Sigma_k^*$ into $\Sigma_m^*$. An example that cannot be obtained from the previous results is also provided.