analyzing functions mapping the unit disk to limaçon domain

This paper focuses on a particular type of starlike univalent function, namely (1 − sz)(1 + z) with −1/3 ≤ s ≤ 1/3. this function maps the open unit disk d onto a limaçon domain denoted by [(u − 1)2 + v2 − s2]2 < (1 − s)2 [(u − 1 − s)2 + v2] . the primary investigation concerns on analytic function families where zh ′ /h maps the unit disk to a subset of this domain. thepaper provides the lower and upper bounds of the real and imaginary parts of these functions and highlights some distinguishing characteristics of these bounds. furthermore, the study employed subordination theory to derive conclusions and corollaries for functionsbelonging to the aforementioned class.