Clunie, J.; Rahman, Q. I. Extension of a theorem of J. H. Grace to transcendental entire functions. (English) Zbl 0776.30020 Math. Proc. Camb. Philos. Soc. 112, No. 3, 565-573 (1992). 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. Reviewer: I.V.Ostrovskii (Khar’kov) Cited in 4 Documents MSC: 30D20 Entire functions of one complex variable (general theory) Keywords:univalent function; Hadamard factorization; normal family PDFBibTeX XMLCite \textit{J. Clunie} and \textit{Q. I. Rahman}, Math. Proc. Camb. Philos. Soc. 112, No. 3, 565--573 (1992; Zbl 0776.30020) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.