Murai 1983 deficiency entire functions fejer gaps

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assertion_p56: Murai's Section 6 shows that an entire function with Fejér gaps takes every complex value infinitely often in any given sector, improving a result of Hayman.

construction_p52: Murai constructs an entire function with Fabry gaps, k/n_k tending to 0, whose Nevanlinna deficiency at 0 equals 1, so his theorem fails when Fejér gaps are weakened to Fabry gaps.

proposition_p46: For an entire function with Fejér gaps and any positive epsilon, the Nevanlinna characteristic is at least (1 - epsilon) times the logarithm of the maximum modulus outside a set of finite logarithmic measure.

theorem_p39: Murai's main theorem: if the exponents of the nonzero Taylor coefficients of an entire function have a convergent reciprocal sum, then every finite value has Nevanlinna deficiency zero.

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Takafumi Murai, The deficiency of entire functions with Fejér gaps. Annales de l'institut Fourier 33 (1983), no. 3, 39-58. doi:10.5802/aif.930. The edition's Numdam cover page prints "© Annales de l’institut Fourier, 1983, tous droits réservés." and "Toute copie ou impression de ce fichier doit contenir la présente mention de copyright." and refers to the Numdam conditions of use (http://www.numdam.org/conditions), every other right reserved.

Murai's main theorem (Section 1, p. 39) states that an entire function with Fejér gaps, meaning that the set $S(f)$ of positive exponents with nonzero coefficients satisfies $\sum1/n_k<\infty$, has no finite deficient value in the sense of Nevanlinna theory. The paper presents it as an improvement of the theorem of Fejér and Biernacki that such a function takes every complex value infinitely often, and of Kövari's theorem on Borel exceptional values. The engine is the Proposition of Section 3 (p. 46): for an entire function with Fejér gaps and any $\epsilon>0$, $m(r,f)\ge(1-\epsilon)\log M(r,f)$ outside a set of finite logarithmic measure, proved through lemmas on trigonometric polynomials (Lemmas 7 and 8) and a reduction (Lemma 9, p. 45) to a Fejér gap series $S$ with $\sqrt r\le\omega(r,S)$ and $\omega(r,S)\le C\Omega(r,S)$ for $r\ge2$. Section 5 (pp. 52--55) shows that the gap hypothesis cannot be weakened to Fabry gaps: it constructs an entire function with Fabry gaps ($k/n_k\to0$) whose deficiency at $0$ is $1$, by alternately taking Taylor polynomials and multiplying by factors $\exp(z/r)^q$ with $q$ large, so that the function behaves locally like $\exp(\eta z)^p$ in the sense of the deficiency. Section 6 (pp. 56--57) uses the same method to show that an entire function with Fejér gaps takes every complex value infinitely often in any given sector.

The introduction (p. 40) cites Clunie's construction of a sequence with $\sum1/n_k=\infty$ such that no entire function with exponents in it has a finite Borel exceptional value, and calls the value distribution of entire functions with $k/n_k\to0$ and $\sum1/n_k=\infty$ difficult to investigate. Problem 517 asks about functions with $n_k/k\to\infty$; for it the summable-reciprocal case is settled by Biernacki's theorem, which the main theorem and the sector assertion strengthen, and the finite-order case by Pólya's theorem, so the part of this regime left open there is the infinite-order part. The Fabry-gap example of Section 5 refutes only the deficiency statement: a deficiency of $1$ at $0$ does not by itself mean that $0$ is taken finitely often, and the paper does not show that it is, so the example is not a counterexample to problem 517.

Read status: claims checked. The Theorem, the Proposition, Lemma 9, the Section 5 construction and assertion (*) were read clause by clause on the printed pages; the proofs were read for their mechanism and not checked step by step.

Source: https://www.numdam.org/item/AIF_1983__33_3_39_0/.

Results

• Theorem (p. 39; proof Section 4, pp. 48--52): an entire function with Fejér gaps has no finite deficient value.
• Proposition (p. 46; proof pp. 46--48): for an entire function with Fejér gaps and $\epsilon>0$, $m(r,f)\ge(1-\epsilon)\log M(r,f)$ outside a set of finite logarithmic measure.
• Section 5 construction (pp. 52--55): an entire function with Fabry gaps whose deficiency at $0$ is $1$.
• Assertion (*) (p. 56; proof pp. 56--57): an entire function with Fejér gaps takes every complex value infinitely often in a given sector.

Bears on. #517: the Theorem and assertion (*) each imply that every entire function with $\sum1/n_k<\infty$ takes every complex value infinitely often, which is the problem's conclusion for that subclass of its hypothesis $n_k/k\to\infty$; the paper says nothing about functions with $n_k/k\to\infty$ and $\sum1/n_k=\infty$, and its Section 5 example does not bear on the problem's question.

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