NDLI: Sharp Bohr Type Real Part Estimates

Content Provider	Springer Nature Link
Author	Kresin, Gershon Maz’ya, Vladimir
Copyright Year	2007
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 $$\left(\sum_{n=m}^\infty \mid c_{n} z^n \mid ^q \right)^{1/q} \leq {2r^{m}\over(1-r^q)^{1/q}} \parallel Re\ f \parallel_{h_{1}}$$ 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 $$\left( {\sum\limits_{n = m}^\infty {\left\| {c_n z^n } \right\|^q } } \right)^{1/q} \leqslant \frac{{2r^m }} {{\left( {1 - r^q } \right)^{1/q} }}\mathop {\sup }\limits_{\left\| \zeta \right\| < 1} \operatorname{Re} \left( {f\left( \zeta \right) - f\left( 0 \right)} \right).$$ As corollary of the above estimate with in the right-hand side, we obtain some sharp Bohr type modulus and real part inequalities.
Starting Page	151
Ending Page	165
Page Count	15
File Format	PDF
ISSN	16179447
Journal	Computational Methods and Function Theory
Volume Number	7
Issue Number	1
e-ISSN	21953724
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-10-26
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Bohr’s inequality Computational Mathematics and Numerical Analysis Functions of a Complex Variable Analysis Inequalities in the complex domain Taylor series Hadamard’s Real Part Theorem Power series
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Theory and Mathematics Analysis

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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