Fekete-Szegő Problems and Toeplitz Determinants for Meromorphic Functions Involving q-Calculus in a Cardioid Domain

Abstract

In this paper, we introduce and investigate a new subclass of meromorphic p-valent functions, denoted by 〖MS〗_(p,q)^* [b;C], defined in the punctured unit disk U^*. The definition of this class is characterized by a subordination condition involving the q-derivative operator and a specific subordinating function, C(z)=1+4/3 z+2/3 z^2, which maps the unit disk onto a Cardioid-shaped domain . We first establish a necessary and sufficient condition for functions to belong to this class using the Hadamard product (convolution). Subsequently, we derive sharp bounds for the first two initial Taylor-Laurent coefficients |a_(1-p) | and |a_(2-p) |. In addition, we also deal with the Fekete-Szegő functional |a_(2-p)-μa_(1-p)^2 | for both real and complex parameters μ. The results obtained in this work generalize several existing findings in the literature and highlight the geometric impact of the Cardioid-shaped domain on meromorphic p-valent starlike functions.