---

Statement

Setting (p. 62). Let

$$f(z)=\sum_{n=0}^\infty a_nz^{\lambda_n}\qquad(1)$$

be an entire (in the paper, integral) function, where $\lambda_n$ is a strictly increasing sequence of non-negative integers. Write $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ for the maximum modulus, $m(r)=\min_{\lvert z\rvert=r}\lvert f(z)\rvert$ for the minimum modulus and $\mu(r)=\max_{n}\lvert a_n\rvert r^{\lambda_n}$ for the maximum term. The conclusion the paper studies is

$$\limsup_{r\to\infty}\frac{m(r)}{M(r)}=\limsup_{r\to\infty}\frac{\mu(r)}{M(r)}=1.\qquad(3)$$

The paper attributes to the last sentence of Pólya (Math. Z. 29 (1929), 549--640) the remark that (3) holds when

$$\liminf_{n\to\infty}\frac{\log(\lambda_{n+1}-\lambda_n)}{\log\lambda_n}>\frac12.\qquad(2)$$

Theorem 1 (p. 62). If

$$\sum_{n=0}^\infty\frac{1}{\lambda_{n+1}-\lambda_n}<\infty,\qquad(4)$$

then (3) holds.

The paper notes (p. 62) that (2) gives $\lambda_{n+1}-\lambda_n>\lambda_n^{1/2+\epsilon}>n^{1+\delta}$ for large $n$ and some positive $\epsilon,\delta$, so (2) implies (4) and Theorem 1 sharpens Pólya's remark. Theorem 1 carries no hypothesis on the order of $f$. Its sharpness is Theorem 2.

Proof pointer

Pp. 64--65, Section 2. An elementary inequality (14)--(15) turns the convergent series $\epsilon_n=1/(\lambda_{n+1}-\lambda_n)$ into a convergent series of block averages $\delta_n$. On the intervals of $\lvert z\rvert$ in which one term $a_kz^{\lambda_k}$ is the maximum term, the paper finds arbitrarily long such intervals, and at their geometric midpoints shows the other terms sum to $o(\lvert a_k\rvert r^{\lambda_k})$, (23). One term then dominates the series on those circles, which gives both ratios in (3).

Read depth

Claims checked: the setting, (2), (3), (4) and Theorem 1 were read clause by clause on the page images of the print, and the proof on pp. 64--65 was followed for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus. The paper's argument is self-contained.

Source. P. Erdős and A. J. Macintyre, Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2) 10 (1954), 62--70; the edition read is named on the source card.

Bears on

• Problem 516: under (4), for every entire $f$ of the form (1) and of any order, the theorem gives $\limsup m(r)/M(r)=1$, which is stronger than the problem's $\limsup\log m(r)/\log M(r)=1$ on the functions it covers. The paper treats only gap condition (4), not the problem's full class; the problem's claim page for this paper records how its finite-order functions fall within that class.