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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The fix-points and factorization of meromorphic functions

Authors: Fred Gross and Chung-chun Yang
Journal: Trans. Amer. Math. Soc. 168 (1972), 211-219
MSC: Primary 30A20
MathSciNet review: 0301175
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Abstract: In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove

(1) If $ f$ and $ g$ are nonlinear entire functions such that $ f(g)$ is transcendental and of finite order, then $ f(g)$ has infinitely many fix-points.

(2) If $ f$ is a polynomial of degree $ \geqq 3$, and $ g$ is an arbitrary transcendental meromorphic function, then $ f(g)$ must have infinitely many fix-points.

(3) Let $ p(z),q(z)$ be any nonconstant polynomials, at least one of which is not $ c$-even, and let $ a$ and $ b$ be any constants with $ a$ or $ b \ne 0$.

Then $ h(z) = q(z)\exp (a{z^2} + bz) + p(z)$ is prime.

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Keywords: Entire functions, meromorphic functions, prime functions, $ E$-pseudo prime, $ E$-prime, fix-points, factorization, differential polynomials, zeros, poles, Nevanlinna theory of meromorphic functions, $ c$-even polynomials
Article copyright: © Copyright 1972 American Mathematical Society