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On the derivatives at the origin of entire harmonic functions

  • D. H. Armitage (a1)

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If f is an entire function in the complex plane such that

where 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).

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References

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1.Armitage, D. H., Uniqueness theorems for harmonic functions which vanish at lattice points, J. Approximation Theory (to appear).
2.Brelot, M. and Choquet, G., Polynômes harmoniques et polyharmoniques, Deuxième colloque sur les équations aux derivées partielles, Bruxelles (1954), 4566.
3.Calderón, A. P. and Zygmund, A., On higher gradients of harmonic functions, Studia Math. 24 (1964), 211226.
4.Hayman, W. K., Power series expansions for harmonic functions, Bull. London Math. Soc. 2 (1970), 152158.
5.Helms, L. L., Introduction to potential theory (New York, 1969).
6.Kuran, Ü., On norms of higher gradients of harmonic functions, J. London Math. Soc. (2) 3 (1971), 761766.
7.Kuran, Ü., On Brelot-Choquet axial polynomials, ibid. (2) 4 (1971), 1526.
8.Müller, C., Spherical harmonics, Lecture Notes in Mathematics, No. 17, (Springer-Verlag, 1966).
9.Pólya, G., Über die kleinsten ganzen Funktionen, deren sämtliche Derivierten im Punkte z=0 ganzzahlig sind, Tôhoku Math. J. 19 (1921), 6568.
10.Straus, E. G., On entire functions with algebraic derivatives at certain algebraic points, Ann. of Math. 52 (1950), 188198.

On the derivatives at the origin of entire harmonic functions

  • D. H. Armitage (a1)

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