Research Problems in Function Theory — Problem 4.3

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. For $N\ge1$, let \[P_N(z)=\sum_{j=1}^Na_jz^{m_j},\qquad 0\le m_1<\cdots<m_N,\quad a_j\in\mathbb C\setminus\{0\}, \] be a polynomial with exactly $N$ nonzero terms, normalized by $\max_{|z|=1}|P_N(z)|\le1$. For $1\le n\le N$, its initial partial sum is $P_n(z)=\sum_{j=1}^na_jz^{m_j}$. Denote by $C(N,n)$ the supremum of $\max_{|z|=1}|P_n(z)|$ over all such polynomials and exponent lists.

Statement recorded in the supplied source.

Determine the sharp bounds $C(N,n)$, and their growth as the number of terms increases. The exponents need not be consecutive: the source fixes a number of terms, rather than imposing a degree bound. [S2]

Short English statement

How large can an initial partial sum of an N-term polynomial be on the unit circle when the full polynomial is bounded there by one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304003 (AMR-022-4003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.3, its complete Update 4.3, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.