Research Problems in Function Theory — Problem 2.59

Definitions and mathematical statement

Let $\rho\ge0$ and $K>0$. Consider a complex power series with coefficients $(a_k)_{k\ge0}$, and write its sections as $S_n(z)=\sum_{k=0}^na_kz^k$ for $n\ge1$. Assume every section is nonzero throughout the same open region \[ W_{\rho,K}=\{x+iy:x>0,\ |y|<Kx^{1-\rho/2}\}. \] In particular no section is identically zero. The width conjecture asks whether this hypothesis forces the series to have infinite radius of convergence, with its sum $f$ satisfying \[ \operatorname{ord}(f)=\limsup_{r\to\infty} \frac{\log\log M(r,f)}{\log r}\le\rho, \qquad M(r,f)=\max_{|z|=r}|f(z)|. \] Polynomials are taken to have order zero. If no convergence is initially assumed, infinite radius is itself part of the conclusion; if the series already converges near zero, the same assertion still includes continuation by that Taylor series to the whole plane. The region has angular width when $\rho=0$ and narrows at the rate dictated by the exponent otherwise. Both $K$ and $\rho$ are independent of $n$. A different zero-free region for each section, or zero-freeness only for a subsequence of sections, is not the stated hypothesis.

Short English statement

Does a common zero-free region of the prescribed width for every Taylor section force an entire sum of order at most the associated exponent?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302059 (AMR-022-2059), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.59. Original source indexed by AMR-022-2059. https://arxiv.org/abs/1809.07200.