On a class of meromorphic \(p\)-valent starlike functions involving certain linear operators (Q1854145)

Summary: Let \(\Sigma_{p}\) be the class of functions \(f(z)\) which are analytic in the punctured disk \({\mathbb E}^*= \{z\in {\mathbb C}: 0< |z|<1\}\). Applying the linear operator \(D^{n+p}\) defined by using the convolutions, the subclass \({\mathcal T}_{n+p}(\alpha)\) of \(\Sigma_{p}\) is considered. The object of the present paper is to prove that \({\mathcal T}_{n+p}(\alpha)\supset {\mathcal T}_{n+p-1}(\alpha)\). Since \({\mathcal T}_{0}(\alpha)\) is the class of meromorphic \(p\)-valent starlike functions of order \(\alpha\), all functions in \({\mathcal T}_{n+p-1}(\alpha)\) are meromorphic \(p\)-valent starlike in the open unit disk \({\mathbb E}\). Further properties preserving integrals and convolution conditions are also considered.