Composite meromorphic functions and normal families |
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Authors: | Wenjun Yuan Bing Xiao Qifeng Wu |
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Affiliation: | (1) The Selmer Center, Department of Informatics, University of Bergen, PB 7800, 5020 Bergen, Norway |
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Abstract: | In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and ({mathcal{F}}) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair ({f(z),g(z)in mathcal{F}}) and one of the following conditions holds: (1) P(z) ? α(z 0) has at least three distinct zeros for any ({z_{0}in D}); (2) There exists ({z_{0}in D}) such that P(z) ? α(z 0) has at most two distinct zeros and α(z) is nonconstant. Assume that β 0 is a zero of P(z) ? α(z 0) with multiplicity p and that the multiplicities l and k of zeros of f(z) ? β 0 and α(z) ? α(z 0) at z 0, respectively, satisfy k ≠ lp, for all ({f(z)inmathcal{F}}). Then ({mathcal{F}}) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion. |
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