## On a new method of analysis and its applications.(English)Zbl 0544.10045

A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XVI, 584 p. £51.00 (1984).
The book is a very nice description of the main results connected with Turán’s power sum theory and its applications to number theory and analysis. Turán’s method is the method of solving different extremal problems similar to the following. Let $$z_ 1,...,z_ n$$ be complex numbers, $$b_ j$$ be fixed numbers, S be a finite set of integers, $$g(\nu)=\sum^{n}_{j=1}b_ jz_ j^{\nu},\quad M(\nu)=\max_{j} | z_ j|^{\nu}.$$ Find $$_{z_ j}\max_{\nu \in S}| g(\nu)| /M(\nu).$$
Turán’s method provides a powerful tool in solving many important problems in number theory and analysis.
Reviewer: G.Kolesnik

### MSC:

 11N30 Turán theory 11-02 Research exposition (monographs, survey articles) pertaining to number theory 30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable 30B10 Power series (including lacunary series) in one complex variable

### Keywords:

power sum theory; extremal problems