The Bohr operator on analytic functions and sections

Abstract The Bohr operator M r for a given analytic function f ( z ) = ∑ n = 0 ∞ a n z n and a fixed z in the unit disk, | z | = r , is given by M r ( f ) = ∑ n = 0 ∞ | a n | | z n | = ∑ n = 0 ∞ | a n | r n . Applying earlier results of Bohr and Rogosinski, the Bohr operator is used to readily establish the following inequalities: if f ( z ) = ∑ n = 0 ∞ a n z n is subordinate (or quasi-subordinate) to h ( z ) = ∑ n = 0 ∞ b n z n in the unit disk, then M r ( f ) ≤ M r ( h ) , 0 ≤ r ≤ 1 / 3 . Further, each k-th section s k ( f ) = a 0 + a 1 z + ⋯ + a k z k satisfies | s k ( f ) | ≤ M r ( s k ( h ) ) , 0 ≤ r ≤ 1 / 2 , and M r ( s k ( f ) ) ≤ M r ( s k ( h ) ) , 0 ≤ r ≤ 1 / 3 . Both constants 1/2 and 1/3 cannot be improved. From these inequalities, a refinement of Bohr's theorem is obtained in the subdisk | z | ≤ 1 / 3 . Also established are growth estimates in the subdisk of radius 1/2 for the k-th section s k ( f ) of analytic functions f subordinate to a concave wedge-mapping. A von Neumann-type inequality is established for the class consisting of Schwarz functions in the unit disk.