Results & Lemmas (6)
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THEOREM 1.
THEOREM 1. Let the nonnegative integer p = p(q, X,Y) be associated with the class Jίq( X, Y) in the following way. (a) Ifq = 0, p = 0. (b)…
THEOREM 1. Let the nonnegative integer p = p(q, X,Y) be associated with the class Jίq( X, Y) in the following way. (a) Ifq = 0, p = 0. (b) Suppose q>\. For each integer m 0, 0 < m 0 < q, consider the system ofq — m0+l equations M L (1.3) Σ a k J e - i k β j - Σ a k J e - i k t > = 0,
Theorem 1
Theorem 1 shows that (1.9) does not hold in general. Suppose, for example, that X = 0,77/180,77/90 and q = 120. Using Theorem 1, we have…
Theorem 1 shows that (1.9) does not hold in general. Suppose, for example, that X = {0,77/180,77/90} and q = 120. Using Theorem 1, we have μ,(l20,3) </>(l20, X, 0 ) = 90 < 120 - 3 + 1. The quantity μe(q, M) has also been studied by E. V. Gleizer. It is my understanding that Gleizer, in a paper [4] submitted to the Ukrainian Journal of Mathematics simultaneously to the submission of this paper, showed Gleizer also obtained a result for entire functions very close to Theorem 1 applied to Sq{X). Th
THEOREM 2.
THEOREM 2. For a > 1 and f ^ q X) of finite order λ, there exists K= K(λ,a,X)>0 such that (1.10) T(ar9f)<KT r,f), r>ro(f).
THEOREM 2. For a > 1 and f $q{X) of finite order λ, there exists K= K(λ,a,X)>0 such that (1.10) T(ar9f)<KT{r,f), r>ro(f).$
Theorem 2
Theorem 2 generalizes to meromorphic functions in many, but not all, cases. A discussion of the possibility of such a generalization…
Theorem 2 generalizes to meromorphic functions in many, but not all, cases. A discussion of the possibility of such a generalization appears in §4. It is elementary that (1.10) implies (Compare to Corollary 2 of [5].) In [12] it is shown that (1.11) implies that the Nevanlinna deficiency is independent of the choice of the origin. From
Theorem 2
Theorem 2 we thus conclude that any entire function of finite order for
Theorem 2 we thus conclude that any entire function of finite order for
Theorem 2
Theorem 2 may fail for the class Jίq(X, Y). For example, let X = θλ where θx = 0 and let 7 = Θ2J3,Θ4,Θ5 where θj = 2τr(j - l)/5, 2 <j < 5.…
Theorem 2 may fail for the class Jίq(X, Y). For example, let X = {θλ} where θx = 0 and let 7 = {Θ2J3,Θ4,Θ5} where θj = 2τr(j - l)/5, 2 <j < 5. Trivially there exist akj > 0 for 1 < j < 5 and all positive integers A: such that (4.1) akιe~^ - Σ akJe- M* = 0. 7 = 2 Suppose q and /„ are arbitrary integers subject only to the condition 1 < q < Jn. By a construction based on our proof of (1.6), we may produce Rn -» oo, βn -> oo, and (4.2, /.W-Π having the following properties: