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Sharp Bohr type real part estimates. (English) Zbl 1152.30002

The authors prove the following main result:
Given an analytic function \(f\) in the unit disk \(\mathbf{D}\) with Re\(f\) in the Hardy space \(h_1(\mathbf{D})\) of harmonic functions on \(\mathbf{D}\), the inequality
\[ (\sum_{n=m}^{+\infty} | c_{n}z^{n}| ^q)^\frac{1}{q}\leq \frac{2r^m}{(1-r^q)^\frac{1}{q}}\| Ref\| _{h_1} \]
holds with the sharp constant, where \(r=| z| <1,m\geq 1.\) This estimate implies sharp inequalities for \(l_q\)-norms of the Taylor series remainder for bounded analytic functions, analytic functions with bounded Re\(f\), analytic functions with bounded Re\(f\) from above, as well as for analytic functions with \(\text{Re}\,f>0\).

MSC:

30A10 Inequalities in the complex plane
30B10 Power series (including lacunary series) in one complex variable
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