The main trends in geometric function theory include the study of convexity, subquadratic functions, and their applications in several variables. These trends contribute to optimization by providing tools for analyzing and solving complex optimization problems. Convexity and subquadratic functions help in understanding the behavior of functions and their optimization properties, which is crucial for developing efficient algorithms. Additionally, geometric function theory aids in the analysis of optimization problems involving inequalities, constraints, and bounds, leading to improved solution methods and better understanding of optimization landscapes. The intersection of geometric function theory with other fields like machine learning and quantum computing opens new avenues for interdisciplinary research and applications in optimization.