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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
DOI: https://doi.org/10.1090/S0002-9947-1972-0301175-2
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, <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$E$">-pseudo prime, <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$E$">-prime, fix-points, factorization, differential polynomials, zeros, poles, Nevanlinna theory of meromorphic functions, <IMG WIDTH="14" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$c$">-even polynomials
Article copyright: © Copyright 1972 American Mathematical Society