Extremal properties of real axially symmetric harmonic functions in

Author:
Peter A. McCoy

Journal:
Proc. Amer. Math. Soc. **67** (1977), 248-252

MSC:
Primary 31B05; Secondary 35C10

DOI:
https://doi.org/10.1090/S0002-9939-1977-0457754-5

MathSciNet review:
0457754

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Abstract | References | Similar Articles | Additional Information

Abstract: The set consists of all real harmonic functions defined by which are regular in , the open unit sphere about the origin . Two problems arise concerning and a subset whose members *U* have the first coefficients specified. (1) For , determine as the limit of a monotone sequence of constants which can be computed algebraically. (2) Find and the constant

**[1]**Richard Askey,*Orthogonal polynomials and special functions*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR**0481145****[2]**Stefan Bergman,*Integral operators in the theory of linear partial differential equations*, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Heft 23, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR**0141880****[3]**P. Dienes,*The Taylor series*, Dover, New York, 1957.**[4]**Robert P. Gilbert,*Constructive methods for elliptic equations*, Lecture Notes in Mathematics, Vol. 365, Springer-Verlag, Berlin-New York, 1974. MR**0447784****[5]**Robert P. Gilbert,*Function theoretic methods in partial differential equations*, Mathematics in Science and Engineering, Vol. 54, Academic Press, New York-London, 1969. MR**0241789****[6]**Ulf Grenander and Gabor Szegö,*Toeplitz forms and their applications*, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR**0094840****[7]**Morris Marden,*Geometry of polynomials*, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR**0225972****[8]**Morris Marden,*Value distribution of harmonic polynomials in several real variables.*, Trans. Amer. Math. Soc.**159**(1971), 137–154. MR**0279323**, https://doi.org/10.1090/S0002-9947-1971-0279323-1**[9]**Peter A. McCoy,*Generalized axisymmetric potentials*, J. Approximation Theory**15**(1975), no. 3, 256–266. MR**0390261****[10]**Peter A. McCoy,*On the zeros of generalized axially symmetric potentials*, Proc. Amer. Math. Soc.**61**(1976), no. 1, 54–58 (1977). MR**0477095**, https://doi.org/10.1090/S0002-9939-1976-0477095-9**[11]**M. Tsuji,*Potential theory in modern function theory*, Maruzen Co., Ltd., Tokyo, 1959. MR**0114894**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0457754-5

Keywords:
Bergman and Gilbert's integral operators,
extremal properties of harmonic functions,
Carathéodory-Fejér theorems

Article copyright:
© Copyright 1977
American Mathematical Society