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Proof of a conjecture of Gross concerning fix-points. (English) Zbl 0681.30014
It is proved that if f and g are transcendental entire functions and if Q is a non-constant polynomial, then the equation f(g(z))=Q(z) has infinitely many solutions. In particular, f(g(z)) has an infinite number of fix-points. This confirms a conjecture of F. Gross [Factorization of meromorphic functions (1972; Zbl 0266.30006)].
Reviewer: W.Bergweiler

30D05Functional equations in the complex domain, iteration and composition of analytic functions
[1]Ahlfors, L.V.: Sur les domaines dans lesquels une fonction méromorphe prend des valeurs appartenant à une région donnée. Acta Soc. Sci. Fennicae (Nova Series A)2, 1–17 (1933)
[2]Bergweiler, W.: An inequality for real functions with applications to function theory. Bull. Lond. Math. Soc.21, 171–175 (1989) · Zbl 0638.30028 · doi:10.1112/blms/21.2.171
[3]Bergweiler, W.: On the fix-points of composite functions. Pac. J. Math. (to appear)
[4]Clunie, J.: The composition of entire and meromorphic functions. In: Shankar, H. (ed.) Mathematical essays dedicated to A. J. Macintyre, pp. 75–92. Athens: Ohio University Press 1970
[5]Dufresnoy, J.: Sur les domaines couvertes par les valeurs d’une fonction méromorphe ou algebroïde, Ann. Sci. Éc. Norm. Supér. III Sér58, 179–259 (1941)
[6]Fatou, P.: Sur l’itération des fonctions transcendantes entières. Acta Math.47, 337–370 (1926) · Zbl 02586475 · doi:10.1007/BF02559517
[7]Gross, F.: Prime entire functions. Trans. Am. Math. Soc.161, 219–233 (1971) · doi:10.1090/S0002-9947-1971-0291456-2
[8]Gross, F.: Factorization of meromorphic functions. Washington, D.C.: U.S. Government Printing Office 1972
[9]Gross, F.: On factorization theory of meromorphic functions. Comment. Math. Univ. St. Pauli24, 47–60 (1975)
[10]Gross, F.: Factorization of meromorphic functions and some open problems. In: Buckholtz, J.D., Suffridge, T.J. (eds.) Complex analysis. nProceedings, Kentucky 1976. (Lect. Notes Math., vol. 599, pp. 51–67) Berlin Heidelberg New York: Springer 1977
[11]Gross, F., Osgood, C.F.: On fixed points of composite entire functions. J. Lond. Math. Soc., II. Ser.28, 57–61 (1983) · Zbl 0508.30024 · doi:10.1112/jlms/s2-28.1.57
[12]Gross, F., Yang, C.C.: The fix-points and factorization of meromorphic functions. Trans. Am. Math. Soc.168, 211–219 (1972) · doi:10.1090/S0002-9947-1972-0301175-2
[13]Hayman, W.K.: Meromorphic functions. Oxford: Clarendon Press 1964
[14]Hayman, W.K.: On the characteristic of functions meromorphic in the plane and of their integrals. Proc. Lond. Math. Soc., III., Ser.14A, 93–128 (1965) · Zbl 0141.07901 · doi:10.1112/plms/s3-14A.1.93
[15]Hayman, W.K.: The local growth of power series: a survey of the Wiman-Valiron method. Can. Math. Bull.17, 317–358 (1974) · Zbl 0314.30021 · doi:10.4153/CMB-1974-064-0
[16]Ozawa, M.: On certain criteria for the left-primeness of entire functions. Kodai Math. Sem. Rep.26, 304–317 (1975) · Zbl 0306.30011 · doi:10.2996/kmj/1138847013
[17]Rosenbloom, P.C.: The fix-points of entire functions. Medd. Lunds Univ. Mat. Sem., M. Riesz, 187–192 (1952)
[18]Valiron, G.: Lectures on the general theory of integral functions. Toulouse: Edouard Privat 1923
[19]Yang, C.C.: On the factorization of entire functions. Ill. J. Math.21, 898–905 (1977)
[20]Yang, C.C.: Progress in factorization theory of entire and meromorphic functions. In: Yang, C.C. (ed.) Factorization theory of meromorphic functions and related topics. (Lect. Notes Pure Appl. Math., vol. 78, pp. 171–192) New York Basel: Dekker 1982
[21]Yang, C.C.: Further results on the fix-points of composite transcendental functions. J. Math. Anal. Appl.90, 259–269 (1982) · Zbl 0498.30030 · doi:10.1016/0022-247X(82)90058-0
[22]Yang, C.C.: On the fix-points of composite transcendental entire functions. J. Math. Anal. Appl.108, 366–370 (1985) · Zbl 0579.30031 · doi:10.1016/0022-247X(85)90031-9