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Transactions of the American Mathematical Society

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On special classes of entire functions whose zeros and growth are restricted


Author: Carl L. Prather
Journal: Trans. Amer. Math. Soc. 258 (1980), 181-189
MSC: Primary 30D15
DOI: https://doi.org/10.1090/S0002-9947-1980-0554327-5
MathSciNet review: 554327
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Abstract: The present paper is an investigation of the uniform limits on bounded sets of entire functions of genus $ \leqslant \,2p$ whose zeros are real, or lie in an even number of sectors (or correspondingly on rays) of a certain size, both the number and size depending on p. A characterization of the uniform limits of entire functions of genus $ \leqslant \,2p$ whose zeros lie in these sectors is given.


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  • [1] S. Hellerstein and J. Korevaar, Limits of entire functions whose zeros and growth are restricted, Duke Math. J. 30 (1963), 221-227. MR 0150304 (27:305)
  • [2] S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227-249. MR 0435393 (55:8353)
  • [3] S. Hellerstein and D. Shea, Minimal deficiencies for entire functions with radially distributed zeros, Proc. London Math. Soc. 37 (1978), 35-55. MR 0486524 (58:6247)
  • [4] J. Korevaar, The zeros of polynomials approximating to an entire function and its canonical representation, thesis, University of Leiden, 1949.
  • [5] -, The zeros of approximating polynomials and the canonical representation of an entire function, Duke Math. J. 18 (1951), 573-592. MR 0043203 (13:222b)
  • [6] -, Limits of polynomials whose zeros lie in a given set, Entire Functions and Related Parts of Analysis, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R. I., 1968, pp. 261-272. MR 0233961 (38:2282)
  • [7] J. Korevaar and C. Loewner, Approximation on an arc by polynomials with restricted zeros, Nederl. Akad. Wetensch. Proc. Ser. A 67 (1964), 121-128. MR 0161984 (28:5186)
  • [8] J. Korevaar and J. E. Lange, Limits of polynomials whose zeros lie in a radial set, Trans. Amer. Math. Soc. 114 (1965), 65-79. MR 0174709 (30:4909)
  • [9] E. Laguerre, Sur les fonctions d genre zero et du genre un, Oeuvres 1 (1898), 174-177.
  • [10] B. Ja. Levin, Distribution of the zeros of entire functions, Trans. Math. Monographs, Vol. 5, Amer. Math. Soc., Providence, R. I., 1964. MR 0156975 (28:217)
  • [11] G. Pólya, Über Annäherung durch Polynome mit Lauter Reelen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), pp. 279-295 = Collected Works, Location of zeros, Vol. 2, M.I.T. Press, Cambridge, Mass., 1974, pp. 54-70.
  • [12] G. Pólya and E. Lindwart, Über einen Zusammenhang zwischen der Konvergenz von Polynomen folgen und der Verteilung ihrer Wurzeln, Rend. Circ. Mat. Palermo 37 (1914), pp. 297-304 = Collected Works, Location of zeros, Vol. 2, M.I.T. Press, Cambridge, Mass. 1974, pp. 54-70.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0554327-5
Keywords: Real entire functions, finite genus, uniform approximation on bounded sets (discs)
Article copyright: © Copyright 1980 American Mathematical Society

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