The Bohr operator on analytic functions and sections

The Bohr operator M-r for a given analytic function f(z) = Sigma(infinity)(n=0) a(n)z(n) and a fixed zin the unit disk, vertical bar z vertical bar = r, is given by M-r (f) = Sigma(infinity)(n=0) vertical bar a(n)vertical bar vertical bar z(n)vertical bar = Sigma(infinity)(n=0) vertical bar a(n)vertical bar r(n). Applying earlier results of Bohr and Rogosinski, the Bohr operator is used to readily establish the following inequalities: if f(z) = Sigma(infinity)(n=0) a(n)z(n) is subordinate (or quasi-subordinate) to h(z) = Sigma(infinity)(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 vertical bar s(k) (f)vertical bar <= 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 vertical bar z vertical bar <= 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. (C) 2020 Elsevier Inc. All rights reserved.