The logarithmic center of a planar region
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- by Douglas Hensley PDF
- Proc. Amer. Math. Soc. 57 (1976), 266-270 Request permission
Abstract:
Given a bounded region $S$ in the complex plane, let $f(\beta ) = {\smallint _S}\log |z - \beta |d$ area for $\beta$ any complex number. A logarithmic center of $S$ is an $\alpha$ which minimizes $f(\beta )$. When is $\alpha$ unique? Conjecture. If $S$ is convex then $\alpha$ is unique. Theorem. If $S$ is convex and symmetric about some line, then $\alpha$ is unique.References
- Douglas Hensley, An asymptotic inequality concerning primes in contours for the case of quadratic number fields, Acta Arith. 28 (1975/76), no. 1, 69–79. MR 382196, DOI 10.4064/aa-28-1-69-79
- I. Kubilyus, The distribution of Gaussian primes in sectors and contours, Leningrad. Gos. Univ. Uč. Zap. Ser. Mat. Nauk 137(19) (1950), 40–52 (Russian). MR 0079610
- Marston Morse and Stewart S. Cairns, Critical point theory in global analysis and differential topology: An introduction, Pure and Applied Mathematics, Vol. 33, Academic Press, New York-London, 1969. MR 0245046
- Tibor Radó, On the problem of Plateau. Subharmonic functions, Springer-Verlag, New York-Heidelberg, 1971. Reprint. MR 0344979
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 57 (1976), 266-270
- MSC: Primary 31A10
- DOI: https://doi.org/10.1090/S0002-9939-1976-0407291-8
- MathSciNet review: 0407291