Abstract
We consider analytic functions ƒ in the unit disk \(\mathbb{D}\) with Taylor coefficients c 0, c 1, … and derive estimates with sharp constants for the l q− norm (quasi-norm for 0 < q < 1) of the remainder of their Taylor series, where q ∈ (0, ∞]. As the main result, we show that given a function ƒ with Re ƒ in the Hardy space \(h_1 \left( \mathbb{D} \right)\) of harmonic functions on \(\mathbb{D}\), the inequality
holds with the sharp constant, where r = ¦z¦ < 1, m ≥ 1. This estimate implies sharp inequalities for l q-norms of the Taylor series remainder for bounded analytic functions, analytic functions with bounded Re ƒ, analytic functions with Re ƒ bounded from above, as well as for analytic functions with Re ƒ > 0. In particular, we prove that
As corollary of the above estimate with in the right-hand side, we obtain some sharp Bohr type modulus and real part inequalities.
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The research of the first author was supported by the KAMEA program of the Ministry Absorption, State of Israel, and by the College of Judea and Samaria, Ariel
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Kresin, G., Maz’ya, V. Sharp Bohr Type Real Part Estimates. Comput. Methods Funct. Theory 7, 151–165 (2007). https://doi.org/10.1007/BF03321638
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DOI: https://doi.org/10.1007/BF03321638