Sharp Real-Part Theorems : A Unified Approach - Gershon Kresin

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Sharp Real-Part Theorems

A Unified Approach

By: Gershon Kresin, T. Shaposhnikova (Editor, Translator)

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Paperback | 2 March 2007

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164 Pages

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22.86 x 15.88 x 1.27

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This volume contains a coherent point of view on various sharp pointwise inequalities for analytic functions in a disk in terms of the real part of the function on the boundary circle or in the disk itself. Inequalities of this type are frequently used in the theory of entire functions and in the analytic number theory.

Industry Reviews

From the reviews: "The subject matter of the book under review is a unified approach to sharp pointwise estimates for analytic functions ! . An index, a list of symbols and 92 references are also provided. Analysts will find this book as an excellent resource for 'real-part theorems' and related inequalities. One can expect rich opportunities for extending the indicated inequalities to analytic functions of several complex variables and solutions of partial differential equations. This work is a welcome addition to the literature." (George Csordas, Zentralblatt MATH, Vol. 1117 (19), 2007)

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Preface	p. V
Introduction	p. XI
Estimates for analytic functions bounded with respect totheir real part	p. 1
Introduction	p. 1
Different proofs of the real-part theorem	p. 2
Extremal values of the real part of the rotated Schwarz kernel	p. 6
Upper estimate of <$>Re { e^{ialpha} Delta f}<$> by the supremum of <$>Re Delta f<$>	p. 8
Two-sided estimates of <$>Re { e^{ialpha} Delta f}<$> by upper and lower bounds of <$>Re Delta f<$>	p. 11
Inequalities for the modulus, real and imaginary parts	p. 12
Variants and extensions	p. 14
Estimates for analytic functions with respect to the L_p-norm of <$>Re Delta f<$> on the circle	p. 17
Introduction	p. 17
Estimate of <$> Re {e^{i alpha} Delta f} <$> by the L_p-norm of <$>Re Delta f<$> on the circle. General case	p. 19
The cases p = 1 and p = 2	p. 23
The case p = ∞	p. 25
Generalization of the Carathéodory and Plemelj inequality	p. 30
Variants and extensions	p. 33
Estimates for analytic functions by the best L_p-approximation of <$>Re f<$> on the circle	p. 37
Introduction	p. 37
Estimate of <$> Re { e^{i alpha} Delta f} <$> by the L_p-norm of <$>Re f<$> - c on the circle. General case	p. 39
The cases p=1 and p = 2	p. 42
The case p = ∞	p. 43
Inequalities for the real and imaginary parts	p. 49
Estimate for the oscillation of <$>Re { e^{i alpha} f}<$> and its corollaries	p. 51
Variants and extensions	p. 53
Estimates for directional derivatives of harmonic functions	p. 57
Introduction	p. 57
Interior estimates for derivatives in a domain	p. 58
Estimates for directional derivatives with constant direction	p. 61
Estimates for directional derivatives with varying direction	p. 63
Estimates for derivatives of analytic functions	p. 69
Introduction	p. 69
Estimate for <$>	p. 73
Estimate for <$>	p. 75
The case p = 1 and its corollaries	p. 77
Explicit estimate in the case p = 1	p. 77
Hadamard's real-part theorem for derivatives	p. 78
Landau type inequality	p. 81
Generalization of the Landau inequality	p. 84
Generalization of the Caratheodory inequality	p. 86
The case p = 2	p. 87
The case p = ∞	p. 92
Variants and extensions	p. 94
Bohr's type real part estimates	p. 99
Introduction	p. 99
Estimate for the l_q-norm of the Taylor series remainder by <$> Re f _1<$>	p. 101
Other estimates for the l_q-norm of the Taylor series remainder	p. 103
Bohr's type theorems	p. 109
Variants and extensions	p. 111
Estimates for the increment of derivatives of analytic functions	p. 115
Introduction	p. 115
Estimate for <$> Delta f^{(n)}(z) <$> by <$> Re { f - cal {P}_m } _p<$>. General case	p. 116
The case p = 1 and its corollaries	p. 118
Explicit estimate in the case p = 1	p. 118
Hadamard-Borel-Carathéodory type inequality for derivatives	p. 119
Landau type inequalities	p. 121
Carathéodory type inequality	p. 124
The cases p =2 and p = ∞	p. 125
References	p. 129
Index	p. 135
List of Symbols	p. 139
Table of Contents provided by Publisher. All Rights Reserved.

Sharp Real-Part Theorems

A Unified Approach

At a Glance

Paperback

Industry Reviews

Shipping

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