A Study of Bohr Phenomenon for Complex Valued and Operator Valued Functions

Abstract

This thesis mainly focuses on the study of Bohr phenomenon for the class of complexvalued newlineanalytic and harmonic functions. The thesis also contains operator-valued analogues newlineof multidimensional Bohr inequality in the complete circular domain. newlineIn Chapter 1, we first give basic definitions of geomeric subclasses of analytic and newlineunivalent functions and discuss some of the geometric and analytic characterization of newlinethese classes. We also discuss several interesting properties of the harmonic mappings in newlinethe complex plane. Furthermore, we give a brief literature survey on the Bohr inequality newlineand Bohr phenomenon. newlineChapter 2 concerns a study of Bohr phenomenon for certain univalent analytic subclasses. newlineWe establish Bohr phenomenon and Bohr inequality for the classes of Ma- newlineMinda type starlike and convex functions, Ma-Minda type starlike and convex functions newlinewith respect to conjugate points, Ma-Minda type starlike and convex functions with newlinerespect to symmetric points, and Ma-Minda type close-to-convex functions. The coefficient newlineestimates for the functions in the aforesaid classes are not yet known. Using newlinesubordination result, we prove Bohr phenomenon and Bohr inequality for the aforesaid newlineclasses. As a consequence, we obtain several interesting corollaries for particular newlinechoices of Ma-Minda functions. newlineIn Chapter 3, we introduce new subclasses of harmonic functions whose analytic newlinepart are Ma-Minda type convex functions and Ma-Minda type convex functions with newlinerespect to conjugate points. We study the growth and area theorem for the aforesaid newlineclasses. Although the coefficient bounds are not known for these classes, using subordination newlineresult, we prove Bohr phenomenon and Bohr inequality for these classes. For newlineparticular choices of Ma-Minda functions, we obtain several interesting corollaries. newlineIn Chapter 4, we extensively study the growth theorem and coefficient estimates for newlinecertain subclasses of harmonic functions in the unit disk D.