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Statement

Setting (p. 62). As in Theorem 1: $f(z)=\sum_{n\ge0}a_nz^{\lambda_n}$ is entire, (1), with $\lambda_n$ a strictly increasing sequence of non-negative integers, and $M(r)$, $m(r)$ and $\mu(r)$ are its maximum modulus, minimum modulus and maximum term.

Theorem 2 (pp. 62--63). If

$$\sum_{n=0}^\infty\frac{1}{\lambda_{n+1}-\lambda_n}=\infty,$$

then there is an entire function of the form (1) such that

$$\limsup_{r\to\infty}\frac{\mu(r)}{M(r)}\le\frac12,\qquad \limsup_{r\to\infty}\frac{m(r)}{M(r)}\le\frac12.\qquad(6)$$

The print numbers the hypothesis (4), the same number it gives the convergent sum of Theorem 1. The paper calls Theorem 1 best possible on the strength of this theorem (p. 62).

Remark on order (pp. 63--64). The paper states that the function constructed for Theorem 2 has finite order if

$$\liminf_{n\to\infty}\frac{1}{\log\lambda_n}\sum_{k=0}^n\frac{1}{\lambda_{k+1}-\lambda_k}>0$$

and zero order if the same quotient tends to $\infty$, and gives this as the reason Theorem 4 cannot be materially strengthened.

Proof pointer

Pp. 65--66, Section 3. The coefficients are chosen by the recursion (24) with the products $A_n$ of (25), built from a sequence $\epsilon_n\to0$ that keeps $\sum\epsilon_n/(\lambda_{n+1}-\lambda_n)$ divergent, so that $a_nr^{\lambda_n}$ is the maximum term exactly for $A_n\le r\le A_{n+1}$ and the next term stays within a factor $e^{-\epsilon_n}$ of it. Two comparable terms give $M(r)>(2-\epsilon)\mu(r)$; choosing the argument of $z$ to set those two terms against each other, or comparing with the mean square $M_2(r)$ in (31)--(33), bounds $m(r)$.

Read depth

Claims checked: Theorem 2, (6) and the order remark were read clause by clause on the page images of the print, and the construction on pp. 65--66 was followed for structure. The order remark is stated in the paper without a separate proof. Nothing here is independently reviewed.

Dependencies

None in the corpus. The construction is the paper's own.

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: the theorem shows that the conclusion $\limsup m(r)/M(r)=1$ of Theorem 1 can fail when the reciprocal gap sum diverges. It bounds the ratio $m(r)/M(r)$, not $\log m(r)/\log M(r)$, so it does not answer the problem's question for any class of functions.