Govorov, N. V. The Riemann boundary value problem with an infinite index. (Kraevaya zadacha Rimana s beskonechnym indeksom). (Russian) Zbl 0632.30001 Moskva: Nauka. 240 p. R. 2.30 (1986). The Riemann problem consists in finding a piecewise analytic function \(\phi\) (z) from the condition \[ \phi^+(t)=G(t)\phi^-(t)+g(t),\quad t\in L, \] where L is a contour on the complex plane which is a boundary of the interior domain \(D^+\) and the exterior domain \(D^-\), \(D^+\cup L\cup D^-=C\), G(t) and g(t) are given functions. The index of the problem is the number \(\kappa:=(2\pi)^{-1}\Delta_ L \arg G\). The well-known book by F. D. Gakhov, “Boundary value problems” (1977; Zbl 0449.30030), contains detailed presentation of the theory of Riemann problems in the case of finite \(\kappa\). In § 49 of this book the results of N. Govorov, D. Berkovich and P. Yurov on the Riemann problem with infinite index are briefly reviewed. The book under review presents a detailed account of the theory of Riemann problems with infinite index. It consists of two parts, each part has 3 chapters. Part I deals with the theory of analytic functions of entirely regular growth in an angle, which is a generalization of Levin- Pfluger theory [see B. Levin, Distributions of zeros of entire functions (1964; Zbl 0152.067)], and a new feature is the influence of the theory of the boundary values of a function being analytic in an angle on the distribution of its zeros in the angle. Part II deals with the Riemann problem with infinite index. In this case the general solution of the Riemann problem with positive infinite index contains, generally speaking, infinitely many linear independent solutions, and the role of polynomials \(P_{\kappa}(z)\) in the case of finite positive \(\kappa\) is played by some analytic functions. The results presented in this monograph are not available elsewhere. In an appendix written by I. V. Ostrovskij the Paley problem is discussed. The Paley’s conjecture was that an entire function f(z) of order \(\rho\) satisfies the inequality \[ \lim (\ln M(r,f)/T(r,f))\leq \pi \rho /\sin (\pi \rho),\quad 0\leq \rho \leq,\quad \leq \pi \rho,\quad \rho >, \] where M and T are defined in Nevanlinna’s theory of meromorphic functions. The first inequality has been proved in 1929 and 1930 by A. Wahlund [Ark. Math. 21 A, N 23, 1-34 (1929)] and G. Valiron (Opuscula Math., A. Wiman dedicata, 1930, pp. 1-12). The second was conjectured by R. E. A. C. Paley [Proc. Cambr. Philos. Soc. 28, 262-265 (1932; Zbl 0005.06703)] and proved by N. V. Govorov [Funct. Anal., Prilozh. 3, No.2, 41-45 (1969; Zbl 0197.053)] and by V. N. Petrenko [Izvest. Akad. Nauk SSSR, Ser. Mat. 33, 414-454 (1969; Zbl 0194.111)]. Reviewer: A.Ramm Cited in 6 ReviewsCited in 40 Documents MSC: 30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable 30E25 Boundary value problems in the complex plane Keywords:Riemann problem; infinite index; Paley’s conjecture Citations:Zbl 0449.30030; Zbl 0152.067; Zbl 0005.06703; Zbl 0197.053; Zbl 0194.111 PDFBibTeX XML