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Extension of a theorem of J. H. Grace to transcendental entire functions. (English) Zbl 0776.30020

The main result of the paper is the following theorem which, in case of polynomials, has been obtained by J. H. Grace in 1902. Let \(f\) be an entire function growing not faster than the order 1 and minimal type. If \(f(1)=f(-1)\) then each of the half-planes \((\text{Re} z\geq 0)\) and \((\text{Re} z\leq 0)\) contains a zero fo \(f'\). Further, if all zeros of \(f'\) are in \((\text{Re} z\geq 0)\) or they are all in \((\text{Re} z\leq 0)\), then these zeros must lie on the imaginary axis. This theorem can be considered as the best possible extension of Grace’s theorem because it ceases to be true if \(f\) is of order \(l\) and normal type.

MSC:

30D20 Entire functions of one complex variable (general theory)
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References:

[1] Grace, Proc. Cambridge Philos. Soc. 11 pp 352– (1902)
[2] DOI: 10.1007/BF01160464 · Zbl 0572.30004 · doi:10.1007/BF01160464
[3] Boas, Entire Functions (1954)
[4] Ahlfors, Complex Analysis (1966)
[5] Heawood, Quart. J. Pure Appl. Math. 38 pp 84– (1907)
[6] Montel, Le?ons sur les Fonctions Univalentes (1933) · JFM 59.0346.14
[7] Marden, Geometry of Polynomials (1966)
[8] Lindel?f, Ann. Sci. ?Ecole Norm. Sup. 22 pp 369– (1905)
[9] DOI: 10.1007/BF01485280 · JFM 48.0082.02 · doi:10.1007/BF01485280
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