Coefficient estimates and determinant inequalities for tilted starlike and convex combination functions associated with the epicycloid domain

This thesis is set within the framework of Geometric Function Theory (GFT), a branch of complex analysis concerned with the geometric properties of analytic functions. In particular, it focuses on the study of subclasses of normalised analytic univalent functions in the open unit disk E ={ z ϵ ℂ : | z | < 1}, where each function f has the form
f (z) = z + a₂z² + a₃z³ + ··· = z + ∑ aₙ z.
The general class of normalised analytic univalent functions is denoted by A . In particular, this research considers two subclasses associated with the epicycloid domain. The first subclass is the tilted starlike subclass, associated with 1 n − cusps epicycloid denoted as Ѕ*т,ₙ₋₁.defined by the subordination condition

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where n≥2 (natural number) and │α│<ϖ₋₂. The second subclass is a convex combination of starlike and convex functions associated with the four-leaf-shaped epicycloid domain, denoted as M ,₄L . It is defined by subordination condition

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where 0 ≤y≤1. This thesis establishes key results including coefficient estimates and coefficient bounds for functions belonging to both classes. The coefficient inequalities for the class Ѕ*т,ₙ₋₁ (a)− and M ,₄L are also discussed which comprise of the upper bounds for the Fekete–Szegö functional, │α ₃-uα₂²│where u can be complex or real numbers and the upper bounds for the second Hankel determinant,│α₂α₄-α₃²│. In addition, this study establishes upper bounds for the Toeplitz determinants Т₂(2), Т₃(1) and Т₃ (2) corresponding to functions in both subclasses. These findings not only extend existing results in the literature but also offer new insights into the geometric characteristics and coefficient structure of analytic functions associated with the epicycloid domains.