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Statement

Setting (p. 39). An entire function $f(z)=\sum c_nz^n$ has Fejér gaps if $S(f)=\{n\ge1;\ c_n\ne0\}$, listed increasingly as $(n_k)$, has $\sum1/n_k<\infty$.

Assertion (*) (Section 6, p. 56). "An entire function with Fejér gaps takes any complex value infinitely often in a given sector."

The proof reduces, without loss of generality, to the sector $\Gamma_\alpha=\{z;\ |\arg z|<\alpha\}$ with $0<\alpha\le1$ (p. 56). The paper presents it as an improvement of Hayman's result (its [8]) that $f$ takes every complex value infinitely often in a given sector if $k(\log k)(\log\log k)^\alpha/n_k=O(1)$ for some $\alpha>2$ (p. 56).

Source. Assertion (*) of Section 6, p. 56, proved on pp. 56--57, 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 statement was read on the printed page. The proof (pp. 56--57) was read for its mechanism and not checked step by step; nothing here is independently reviewed.

Proof pointer

Section 6 (pp. 56--57). For lower order $\rho(f)<\infty$ the paper cites Hayman [8]; for $\rho(f)=\infty$ it reduces to the value $0$, the sector $\Gamma_\alpha=\{|\arg z|<\alpha\}$ with $0<\alpha\le1$, and $f(0)=1$, and argues by contradiction. If $f$ has finitely many zeros in $\Gamma_\alpha$, a Green's-function count of zeros in truncated sectors is $O(1)$ (38). Bounds on the normal derivative of Green's function (39), from Petrenko [12], give a lower bound (40) for that count. The Proposition makes its main term at least $(\alpha\eta/8)\log M(r)$ log-finely, and the argument of Section 4 makes a negative term $o(\log M(r))$. Infinite lower order then supplies a set of infinite logarithmic measure on which the remaining terms are small, so the count tends to infinity along it, contradicting (38) (p. 57).

Dependencies

The Proposition of Section 3; the method of Section 4; Hayman [8] for finite lower order; Green's-function estimates from Petrenko [12].

Bears on

• Problem 517: settles the instances with $\sum1/n_k<\infty$ directly, in the stronger form that every value is taken infinitely often in every sector. It says nothing about exponents with $\sum1/n_k=\infty$.