Subordination Criteria for Meromorphic m-Valent Functions with Balloon-Shaped Domains

In this paper, we introduce two new subclasses of meromorphic m-valent functions in the punctured unit disk \mathbb{U}^*, denoted by \mathscr{M}\mathscr{S}^*_{p,q}(bal) and \mathscr{M}\mathscr{C}_{p,q}(bal). These classes are defined via (p,q)-derivative operator and a subordination condition involving the balloon-shaped function\[\mathcal{B}(\mathfrak{z})=\frac{۲\sqrt{۱+\mathfrak{z}}}{۱+e^{-\mathfrak{z}}}.\]Necessary and sufficient conditions for inclusion in these subclasses are established, extending several known results for meromorphic multivalent functions.

کلیدواژه ها:

m-valent functions ، Balloon-shaped domain ، Subordination ، (P ، q)-derivative operator