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Statement

Setting (pp. 39--41). 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 $\sum1/n_k<\infty$ (p. 39). $M(r,f)$ is the maximum modulus on $|z|=r$ and $m(r,f)=(1/2\pi)\int_0^{2\pi}\log^+|f(re^{it})|\,dt$ (p. 40). A set $E\subset(0,\infty)$ has finite logarithmic measure if $\int_E dr/(1+r)<\infty$, and $A(r)\le B(r)$ holds log-finely (l.f.) if it holds outside such a set (p. 41).

Proposition (Section 3.1, p. 46). Let $f$ be an entire function with Fejér gaps and let $\epsilon>0$. Then

$$m(r,f)\ge(1-\epsilon)\log M(r,f)\qquad\text{(l.f.)},$$

the paper's inequality (11). The paper calls it "interesting in itself" (p. 46).

Source. The Proposition of Section 3, p. 46, 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 and the definitions it uses were read clause by clause on the printed pages. The proof (pp. 46--48) was read for its mechanism and not checked step by step; nothing here is independently reviewed.

Proof pointer

Section 3 (pp. 46--48). Lemma 9 (p. 45) lets one assume the exponent set $S$ satisfies $\sqrt r\le\omega(r,S)$ and $\omega(r,S)\le C\Omega(r,S)$ for $r\ge2$, where $\omega(r,S)$ counts the $n_k<r$ and $\Omega(r,S)=\int_0^r\omega(x,S)\,dx/x$; normalize $a_0=1$. Lemma 10 (p. 47) shows that the tail of the series beyond a cut-off $u_r$, chosen so that a majorant of $\Omega$ at $u_r$ equals $5\log\mu(r,f)$, is at most $1$ log-finely. The truncated polynomial has at most about $\omega(u_r)$ terms, so Lemma 8 (p. 44), $m(P)\ge\log^+\max|\hat P(k)|-Cn$ for a trigonometric polynomial with $n$ nonzero coefficients, together with Wiman's Lemma 3 (p. 42) on the maximum term, gives $m(P_r)\ge(1-o(1))\log M(r)$ log-finely (15). A measure count of the set where $\log^+|P_r|$ is large then transfers the bound to $f$ (p. 48).

Dependencies

Lemmas 3, 5, 8, 9 and 10 of the paper (pp. 42--47); Lemma 3 is Wiman's theorem (the paper's [15]) and Lemma 8 rests on Lemma 7 (p. 43).

Bears on

• Problem 517: indirect. It is the main analytic input to the Theorem and to the sector assertion, which settle the instances with $\sum1/n_k<\infty$; it says nothing about value distribution by itself.