الفهرس 
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Abstract
Complex analysis is one of the fields of pure mathematics. Geometric function theory (GFT) is a fundamental branch of mathematics that focuses on the geometric features of analytic functions. Contributors in this field include Riemann, Cauchy, Weierstrass, and Koebe. The Riemann mapping theorem allows us to replace any arbitrary domain G ⊂ C with at least two border points with the open unit disk D. Koebe’s conclusion states that if analytic functions have the further feature of being univalent in D, then the confirmation of conformality and the assertion of Riemann mapping theorem are verified. The primary objective of this thesis is to investigate the geometrical characteristics of particular analytic functions whose importance is clear throughout relevant applications. The present thesis consists of four chapters, illustrated as follows: Chapter 1 introduce the notation and standard definitions associated with the GFT. Moreover, it highlights many families of analytic functions and offers historical insights into the corresponding investigations. Chapter 2 introduces a short survey of Bernoulli’s numbers and their connection to the well-known Einstein functions. The application of the q-difference operator yields a set of extensive families of bi-univalent analytic functions that are subordinate to modified Einstein functions. We also investigate methods for determining the upper bound value of the initial Maclaurin coefficients and the Fekete-Szego inequalities for functions within these families. At the conclusion of this chapter, the highlights of the published results that were extracted from this chapter are provided. Chapter 3 uses the permissible function approach to figure out what conditions must be met for the normalized Galue-type Struve function to be starlike, univalent, and convex. Furthermore, we provide the criteria that determine when these functions can be considered close-to-convex according to certain convex functions.