The Bohr operator on analytic functions and sections. (English) Zbl 1461.30006
Summary: The Bohr operator \(\mathcal{M}_r\) for a given analytic function \(f(z) = \sum_{n = 0}^\infty a_n z^n\) and a fixed \(z\) in the unit disk, \(| z | = r\), is given by \[\mathcal{M}_r(f) = \sum_{n = 0}^\infty | a_n | | z^n | = \sum_{n = 0}^\infty | a_n | r^n.\] Applying earlier results of Bohr and Rogosinski, the Bohr operator is used to readily establish the following inequalities: if \(f(z) = \sum_{n = 0}^\infty a_n z^n\) is subordinate (or quasi-subordinate) to \(h(z) = \sum_{n = 0}^\infty b_n z^n\) in the unit disk, then \[\mathcal{M}_r(f) \leq \mathcal{M}_r(h), \quad 0 \leq r \leq 1 / 3.\] Further, each \(k\)-th section \(s_k(f) = a_0 + a_1 z + \cdots + a_k z^k\) satisfies \[| s_k ( f ) | \leq \mathcal{M}_r ( s_k ( h ) ), \quad 0 \leq r \leq 1 / 2,\] and \[\mathcal{M}_r ( s_k ( f ) ) \leq \mathcal{M}_r( s_k(h)), \quad 0 \leq r \leq 1 / 3.\] Both constants 1/2 and 1/3 cannot be improved. From these inequalities, a refinement of Bohr’s theorem is obtained in the subdisk \(| z | \leq 1 / 3\). 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.
MSC:
| 30B10 | Power series (including lacunary series) in one complex variable |
| 30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |
| 30A10 | Inequalities in the complex plane |
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