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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

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 34M99 Ordinary differential equations in the complex domain
##### Keywords:
differential polynomial; fix-points