Bohr’s Inequality And Its Extensions - Ng, Zhen Chuan (2017)
This thesis focuses on generalizing the Bohr’s theorem. Let h be a univalent function defined on U. Also, let R(a; g;h) be the class of functions f analytic in U such that the differential f (z)+az f 0(z)+gz2 f 00(z) is subordinate to h(z). The Bohr’s theorems for the class R(a; g;h) are proved for h being a convex function and a starlike function with respect to h(0). The Bohr’s theorems for the class of analytic functions mapping U into concave wedges and punctured unit disk are next obtained in the following chapter. The classical Bohr radius 1=3 is shown to be invariant by replacing the Euclidean distance d with either the spherical chordal distance or the distance in Poincaré disk model. Also, the Bohr’s theorem for any Euclidean convex set is shown to have its analogous version in the Poincaré disk model. Finally, the Bohr’s theorems are obtained for some subclasses of harmonic and logharmonic mappings defined on the unit disk U.