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Sharp real-part theorems. A unified approach. Translated from Russian and edited by T. Shaposhnikova. (English) Zbl 1117.30001
Lecture Notes in Mathematics 1903. Berlin: Springer (ISBN 978-3-540-69573-8/pbk). xvi, 140 p. EUR 29.95/net; SFR 49.50; $ 44.95; £ 23.00 (2007).
Estimates for analytic functions and their derivatives play an important role in complex analysis and its applications. The subject matter of the book under review is a unified approach to sharp pointwise estimates for analytic functions, $f(z)$, in a disk $D(0,R):=\{ z\in \Bbb C\,:\, \vert z\vert < R\}$, in terms of the real part of $f(z)$ on the boundary circle. Inequalities with sharp constants are obtained from the analysis of Schwarz integral representation of $f(z)$. The book contains seven chapters. The chapter headings are as follows: 1. Estimates for analytic functions bounded with respect to their real part, 2. Estimates for analytic functions with respect to the $L_p$-norm of   $\text{Re}(f(z)-f(0))$ on the circle, 3. Estimates for analytic functions by the best $L_p$-approximation of $\text{Re }f$ on the circle, 4. Estimates for directional derivatives of harmonic functions, 5. Estimates for derivatives of analytic functions, 6. Bohr’s type real part estimates, 7. Estimates for the increment of derivatives of 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.

30-01Textbooks (functions of one complex variable)
30A10Inequalities in the complex domain
30B10Power series (one complex variable)
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
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