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Statement

Setting (pp. 39--41). A Fejér gap series is a sequence $n_1<n_2<\cdots$ of positive integers with $\sum_{k=1}^\infty1/n_k<\infty$. An entire function $f(z)=\sum_{n=0}^\infty c_nz^n$ has Fejér gaps if $S(f)=\{n\ge1;\ c_n\ne0\}$ is a Fejér gap series; the constant term is not restricted (p. 39). For an entire $g$ the paper writes $M(r,g)=\max\{|g(z)|;\ |z|=r\}$,

$$m(r,g)=\frac1{2\pi}\int_0^{2\pi}\log^+|g(re^{it})|\,dt$$

for the characteristic function, $N(r,a,g)=\int_0^r n(x,a,g)\,dx/x$ with $n(x,a,g)$ the number of roots of $g(z)=a$ in $0<|z|<x$ counted with multiplicity, and defines the deficiency at $a\in\mathbb C$ by

$$\delta(a,g)=1-\limsup_{r\to\infty}\frac{N(r,a,g)}{m(r,g)}$$

(pp. 40--41). A finite deficient value is an $a\in\mathbb C$ with $\delta(a,g)>0$.

Theorem (p. 39). "An entire function with Fejér gaps has no finite deficient value."

That is, if $f$ has Fejér gaps then $\delta(a,f)=0$ for every $a\in\mathbb C$. The paper presents this as an improvement of the theorem of Fejér and Biernacki (its [4] and [1]) that an entire function with Fejér gaps takes every complex value infinitely often, and of Kövari's theorem (its [9]) that an entire function has no finite Borel exceptional value when $S(f)=(n_k)$ satisfies $\lim_{k\to\infty}n_k\eta(k)/\log\log k=\infty$ for some positive increasing $\eta$ on $(0,\infty)$ with $\int_0^\infty\eta(r)\,dr<\infty$, both conditions as printed (p. 39). It notes (p. 40) that the bound $\Delta(f)\le C\rho(f)D(f)$ of its [6, 11] already gives the theorem for functions of finite lower order, since Fejér gaps force $D(f)=0$, a remark it credits to Fuchs; the theorem is new information when the lower order is infinite.

The hypothesis cannot be weakened to Fabry gaps: Section 5 builds an entire function with Fabry gaps and $\delta(0,\cdot)=1$.

Source. The Theorem of Section 1, p. 39, of Takafumi Murai, The deficiency of entire functions with Fejér gaps, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 39--58, doi:10.5802/aif.930, as identified on the source card.

Read depth. Claims checked: the definitions (pp. 39--41) and the statement were read clause by clause on the printed pages. The proof of Section 4 (pp. 48--52) was read for its mechanism and not checked step by step; nothing here is independently reviewed.

Proof pointer

Section 4 (pp. 48--52). By Lemma 9 the exponent set is enlarged to a Fejér gap series satisfying the regularity condition (9), and the paper notes it suffices to prove $\delta(0,f)=0$ with $f(0)=1$ (p. 48). Lemma 11 (p. 48) shows that, log-finely, the maximum of $|f|$ on every short arc of a slightly larger circle is at least $\exp\{-C_0\Omega(u_r)\}$; the proof integrates against a convolution of triangular kernels whose Fourier transform vanishes at the exponents up to the cut-off. With the Proposition this gives the estimates (18), and Lemma 12 (p. 50) reduces the theorem to $m(\tilde r,1/g_r)=o(m(r))$ outside a set of finite logarithmic measure, where $g_r$ is $f$ with its zeros in an annulus replaced by their reflections through a circle, as in (19), and multiplied by the constant $\exp\{C_0\Omega(u_r)\}$. That bound is proved in 4.2 (pp. 50--52) by integrating $\partial_t\log|g_r|$ (Lemma 2) over the short arcs where $|g_r|<1$.

Dependencies

The Proposition of Section 3; Lemmas 1, 2 and 5 (pp. 41--42), Lemma 9 (p. 45) and Lemmas 10--12 (pp. 47--50) of the paper; Lemmas 1 and 2 are cited to Hayman's Meromorphic functions.

Bears on

• Problem 517: settles the instances with $\sum1/n_k<\infty$. The theorem itself concerns deficiencies; the step to infinitely many $a$-points is the standard one that a transcendental entire function taking a value only finitely often has deficiency $1$ there. The paper does not treat functions with $n_k/k\to\infty$ and $\sum1/n_k=\infty$, which its introduction calls difficult (p. 40).