On the Study of Starlike Functions Associated with the Generalized Sine Hyperbolic Function

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since ℜ ( 1 + sinh ( z ) ) ≯ 0 , it implies that the class S sinh * introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter λ with the restriction 0 ≤ λ ≤ ln ( 1 + 2 ) , and by doing that, ℜ ( 1 + sinh ( λ z ) ) > 0 . The present research intends to provide a novel subclass of starlike functions in the open unit disk U , denoted as S sinh λ * , and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients a n for n = 2 , 3 , 4 , 5 . Then, we prove a lemma, in which the largest disk contained in the image domain of q 0 ( z ) = 1 + sinh ( λ z ) and the smallest disk containing q 0 ( U ) are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including S * ( β ) and K ( β ) of starlike functions of order β and convex functions of order β . Investigating S sinh λ * radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding S sinh λ * radii of different subclasses is the calculation of that value of the radius r < 1 for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of λ , is also obtained.