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The deficiency of a gap series. (English) Zbl 0533.30028
Let f be an entire function of finite lower order \(\lambda\) which has power series expansion \((*)\quad f(z)=\sum^{\infty}_{k=0}a_ kz^{n_ k}.\) If \(\quad 0<D\leq \pi\) and \(\{e^{i_{n_ kt}}\}^{\infty}\!_{k=k_ 0}\) is incomplete in \(L^ 2(-D,D)\) for some \(k_ 0\geq 1\), then \(\sum_{a\in {\mathbb{C}}}\delta(a,f)\leq K(\lambda D)^{3/2}\) where K is a constant and \(\delta\) (a,f) is the Nevanlinna deficiency of f at a. A corollary to the theorem is that if f is an entire function with lower order \(\lambda\) and of a form (*) such that \(\lim \inf_{k\to \infty}(n_{k+1}-n_ k)=q,\) then \(\sum_{a\in {\mathbb{C}}}\delta(a,f)\leq C(\lambda /q)^{3/2}\) where C is an absolute constant. These results affirm a conjecture of W. H. J. Fuchs [Sympos. theor. Phys. Math. 9, Lectures 1968 sixth Annivers. Sympos. Inst. math. sci. Madras, India 177-181 (1969; Zbl 0179.110)] who had earlier theorems of this nature. The proof of the theorem exhibits a considerable amount of technical expertise.
Reviewer: L.R.Sons

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30B10 Power series (including lacunary series) in one complex variable
30D10 Representations of entire functions of one complex variable by series and integrals
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