Schlicht Taylor series converging uniformly but not absolutely on the unit circle: must the exponents satisfy more than $\sum 1/m_k=\infty$ (Erdős–Herzog–Piranian, 1951)?

The 1951 paper of Erdős, Herzog and Piranian studies Taylor series $\sum_{k=0}^\infty a_k z^{m_k}$ that represent a schlicht (univalent) function in the unit disc and converge uniformly, but not absolutely, on the unit circle. From Landau's coefficient bound $\sum m_k|a_k|^2<\infty$ and Schwarz's inequality they derive the necessary condition $\sum_{k=0}^\infty 1/m_k=\infty$ on the exponent sequence, and they construct two examples of the phenomenon, one with a single boundary singular point and one whose exponents have Fabry gaps ($m_{k+1}-m_k\to\infty$). They then ask, as an open question, whether these hypotheses force a restriction on the exponents that is strictly stronger than this divergence condition. The exponent question remains open.

Background

For a power series of radius of convergence 1, convergence on the unit circle $C:\lvert z\rvert=1$ can be strong or weak. The series converges absolutely on $C$ if $\sum_n\lvert c_n\rvert<\infty$, and uniformly on $C$ if the partial sums converge uniformly in the boundary point. Hardy (1913) proved that uniform convergence on the circle of convergence does not force absolute convergence, by exhibiting (non-univalent) series that converge uniformly but not absolutely on $|z|=1$.

A function analytic in the open unit disc is called schlicht (the term of the classical German literature; the modern term is univalent) if it takes every value at most once, so that it maps the disc conformally one-to-one onto its image. For such functions the boundary behavior of the Taylor series is constrained: Fejér's theorem (1913) states that if a function is schlicht in the open unit disc and continuous on the closed disc, then its Taylor series converges uniformly on the closed disc. Erdős–Herzog–Piranian (1951) exhibited two functions, each schlicht on the closed unit disc, whose Taylor series converge uniformly but not absolutely on $C$; one of them has only a single singular point on $C$, while the Taylor series of the other, written as $\sum_{k\ge0}a_k z^{m_k}$ with $m_0<m_1<\cdots$ the indices of the nonzero coefficients, has Fabry gaps: $m_{k+1}-m_k\to\infty$. (A series most of whose coefficients vanish is called lacunary; Fabry gaps and Hadamard gaps — the latter meaning $m_{k+1}/m_k\ge q>1$ for some fixed $q>1$ — are the two classical gap structures, and the gap theorems of Hadamard and Fabry assert that, for series of radius 1 whose coefficients are not exponentially small, such gaps force the circle of convergence to be a natural boundary, that is, the function admits no analytic continuation across any point of it.)

The paper also records the known necessary arithmetic restriction on the exponents. By a coefficient theorem of Landau (1929, p. 65, as cited in the paper), if the series represents a schlicht function and converges uniformly but not absolutely on $C$, then $\sum_{k=0}^\infty m_k\lvert a_k\rvert^2<\infty$; since the convergence is not absolute, $\sum_k\lvert a_k\rvert=\infty$, and Schwarz's inequality applied to $\lvert a_k\rvert=\sqrt{m_k}\,\lvert a_k\rvert\cdot m_k^{-1/2}$ gives

$$\Big(\sum_{k=0}^\infty\lvert a_k\rvert\Big)^{2}\le\Big(\sum_{k=0}^\infty m_k\lvert a_k\rvert^2\Big)\Big(\sum_{k=0}^\infty 1/m_k\Big),$$

so that the exponent sequence must satisfy condition (2) of the paper:

$$\sum_{k=0}^{\infty}\frac{1}{m_k}=\infty .$$

Since Hadamard gaps imply $\sum 1/m_k<\infty$, condition (2) already rules out Hadamard-lacunary exponent sequences in this class, while Fabry-gap sequences with $\sum 1/m_k=\infty$ (for instance exponents growing like $k\log k$) survive it — and the paper's second example realizes exactly such a sequence. The summary of the paper closes with the question: "It remains an open question whether the condition implies a restriction on {m_k} which is stronger than (2)." Here 'the condition' is the paper's full hypothesis — that the series represent a schlicht function and converge uniformly but not absolutely on $C$ — and the question is whether the class of exponent sequences that can occur for such series is strictly smaller than the class allowed by (2).

Problem Statement

The problem is the open question stated at the end of §1 of Erdős–Herzog–Piranian (Pacific J. Math. 1 (1951), 75–82): "It remains an open question whether the condition implies a restriction on {m_k} which is stronger than (2)."

Setting: let $\{m_k\}_{k\ge0}$ be a strictly increasing sequence of positive integers, and let $f(z)=\sum_{k=0}^\infty a_k z^{m_k}$ with $a_k\ne0$ be a Taylor series, $m_k$ being the indices of the nonzero coefficients. (If the series has a nonzero constant term, that single term is absolutely convergent and irrelevant to every convergence and univalence condition below, so it is dropped from the enumeration and the exponents are taken positive, $m_k\ge1$; this is the convention under which the paper's condition (2), involving $1/m_k$, is stated.) Consider the hypotheses: (i) $f$ represents a schlicht (univalent) function in the open unit disc; (ii) the series converges uniformly on the unit circle $C:\lvert z\rvert=1$, that is, the partial sums $\sum_{k\le K}a_k z^{m_k}$ converge uniformly over all $z\in C$; (iii) the series does not converge absolutely, that is, $\sum_{k=0}^\infty\lvert a_k\rvert=\infty$. Erdős–Herzog–Piranian proved that (i)–(iii) imply $\sum_{k=0}^\infty 1/m_k=\infty$ (their condition (2)), and their two examples — functions schlicht even on the closed unit disc, the second with Fabry-gap exponents — show the class is nonempty.

Question: do the hypotheses (i)–(iii) imply a necessary restriction on the exponent sequence $\{m_k\}$ that is strictly stronger than $\sum_{k=0}^\infty 1/m_k=\infty$? Here a 'restriction stronger than (2)' means a property $R$ of strictly increasing sequences of positive integers such that (a) every series satisfying (i)–(iii) has an exponent sequence with property $R$, and (b) $R$ is strictly stronger than condition (2), that is, at least one strictly increasing sequence of positive integers with $\sum_{k=0}^\infty 1/m_k=\infty$ fails $R$.

Equivalently, the question asks whether a strictly stronger necessary condition on the exponents is forced by (i)–(iii), or whether condition (2) is sharp for this class — that is, whether for every strictly increasing sequence of positive integers $\{m_k\}$ with $\sum_{k=0}^\infty 1/m_k=\infty$ there exist coefficients $a_k\ne0$ such that $\sum_k a_k z^{m_k}$ satisfies (i)–(iii). The question is posed in the source's original generality: the hypotheses are exactly (i)–(iii), with no auxiliary restriction (boundedness of the image, gap structure, normalization, or special boundary geometry), and $R$ is a property of the exponent sequence $\{m_k\}$ itself (a restriction on the coefficients $a_k$ alone is not a candidate for $R$).

References

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