Symmetry and uniform approximation by analytic functions

Author:
Dmitry Khavinson

Journal:
Proc. Amer. Math. Soc. **101** (1987), 475-483

MSC:
Primary 30E10

DOI:
https://doi.org/10.1090/S0002-9939-1987-0908652-8

MathSciNet review:
908652

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we treat the problem of finding all the domains in for which the uniform distance from the function to the space of analytic functions is equal precisely to (2 area/perimeter). We show that for simply connected domains it occurs if and only if the domain is a disk. We also discuss the relation of the above problem to certain types of symmetry in potential theory and to the theory of Schwarz functions.

**[1]**D. Aharanov and H. S. Shapiro,*Domains on which analytic functions satisfy quadrature identities*, J. Analyse Math.**30**(1976), 39-73. MR**0447589 (56:5899)****[2]**H. Alexander,*Projections of polynomial hulls*, J. Funct. Anal.**3**(1973), 13-19. MR**0338444 (49:3209)****[3]**-,*On the area of the spectrum of an element of a uniform algebra*, Complex Approximation (B. Aupetit, ed.), Birkhäuser, Basel, 1980, pp. 3-12. MR**578634 (82b:32015)****[4]**P. J. Davis,*The Schwarz function and its applications*, Carus Math. Monographs, vol. 17, (1974). MR**0407252 (53:11031)****[5]**P. Duren,*Theory of**-spaces*, Academic Press, New York and London, 1970. MR**0268655 (42:3552)****[6]**T. Gamelin and D. Khavinson,*The isoperimetric inequality and rational approximation*(in preparation).**[7]**P. Garabedian,*On the shape of electrified droplets*, Comm. Pure Appl. Math.**28**(1965), 31-34. MR**0175491 (30:5675)****[8]**E. L. Ince,*Ordinary differential equations*, Longmans, Green and Co., London, New York and Toronto, 1927.**[9]**D. Khavinson,*Annihilating measures of the algebra*, J. Funct. Anal.**28**(1984), 175-193. MR**757994 (86d:46048)****[10]**-,*A note on Toeplitz operators*, Geometry of Banach Spaces (N. Kalton and E. Saab, eds.), Lecture Notes in Math., vol. 934, Springer-Verlag, Berlin and New York, 1986, pp. 89-95. MR**827763****[11]**-,*Smirnov classes of analytic functions in multiply connected domains*, Appendix to English translation of*Foundations of the theory of extremal problems for bounded analytic functions and their various generalizations*, by S. Ya. Khavinson, Amer. Math. Soc. Transl.**129**(1986), 57-61.**[12]**D. Khavinson and D. Luecking,*On an extremal problem in the theory of rational approximation*, J. Approx. Theory (to appear). MR**888294 (88g:41013)****[13]**S. Ya. Khavinson,*Foundations of the theory of extremal problems for bounded analytic functions and their various generalizations*, Amer. Math. Soc. Transl.**129**(1986), 1-57.**[14]**S. Ya. Khavinson and G. C. Tumarkin,*On the definition of analytic functions of class**in multiply connected domains*, Uspekhi Mat. Nauk**13**(1958), 201-206. (Russian) MR**0093590 (20:114)****[15]**A. A. Kosmodem' yanskii,*A converse of the mean value theorem for harmonic functions*, Russian Math. Surveys**36**(1981), no. 5, 159-160; translated from Russian. MR**637445 (84d:31001)****[16]**M. Sakai,*Quadrature domains*, Lecture Notes in Math., vol. 934, Springer-Verlag, Berlin and New York, 1982. MR**663007 (84h:41047)****[17]**J. Serrin,*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.**43**(1971), 304-318. MR**0333220 (48:11545)****[18]**H. Weinberger,*Remark on the preceeding paper of Serrin*, Arch. Rational Mech. Anal.**43**(1971), 319-320. MR**0333221 (48:11546)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30E10

Retrieve articles in all journals with MSC: 30E10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0908652-8

Article copyright:
© Copyright 1987
American Mathematical Society