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On differential polynomials. (English) Zbl 0643.30022
Analysis of one complex variable, Proc. Am. Math. Soc. 821st Meet., Laramie/Wyo. 1985, 12-32 (1987).
[For the entire collection see Zbl 0634.00005.]
A differential polynomial P(z) in a given meromorphic function f is a finite sum of expressions \[ \Phi_ k(z) = a_ k(z)\prod^{p}_{j=0}(f^{(j)})^{s_{k_ j}}, \] where the coefficients \(a_ k\) are meromorphic in the plane and satisfy \(T(r,a_ k)=S(r,f)\). The numbers \(\bar d(P)\) and \(\underline{d}(P)\) are defined to be the maximum and minimum values of \(\sum^{p}_{j=0}s_{k_ j}\) (1\(\leq k\leq n)\). The main results of the present paper are:
Theorem 1: \[ T(r,f)\leq \frac{1}{\underline d(f)}m(r,\frac{1}{p})+N(r,\frac{1}{f})+S(r,f),\;if\;\underline d(P)>0. \] Theorem 2: \[ T(r,f)\leq \frac{\underline d(f)}{\underline d(f)- 1}N(r,\frac{1}{f})+\frac{1}{\underline d(f)-1}\bar N(r,\frac{1}{P- 1})+S(r,f),\;if\;\underline d(P)>1. \] The second inequality may be viewed as a generalization of a result of W. K. Hayman [Theorem 3.5 in: Meromorphic functions (1964; Zbl 0115.062)] but it does not contain it. Several applications to differential equations and fix-points of meromorphic functions are given.
Reviewer: N.Steinmetz

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M99 Ordinary differential equations in the complex domain