Executive Summary : | Geometric Function Theory is a branch of complex analysis that focuses on the geometric properties of analytic functions and harmonic mappings. Founded around the turn of the 20th century, it remains an active field of research. Despite the famous coefficient problem, the Bieberbach conjecture, solved by Louis de Branges in 1984, it suggests various approaches and directions for the study of geometric function theory. Recent developments include a correspondence between ingredients in geometric function theory and the non-perturbative crossing symmetric representation of 2-2 scattering amplitudes of identical scalars. The theory also includes the Riemann mapping theorem, Maximum principle, Schwarz's Lemma, and Riemann-Hurwitz formula. Complex-valued functions have interesting geometric properties such as starlikness, convexity, close-to-convexity, and close-to-starlikeness. The study of Geometric Function Theory is an interesting topic, with several conjectures attracted to modern function researchers. The current problems studied include finding sharp bounds of logarithmic coefficients, establishing sharp Bohr-type inequalities and the corresponding Bohr radius, finding sharp bounds of Hankel determinants, finding variability regions of a certain class of functions precisely, and addressing long-standing conjectures like the Bieberbach conjecture and Zalcman conjecture. Therefore, continuing these studies for certain geometric classes of functions is necessary to develop theories to solve problems and conjectures. |