Research Problems in Function Theory — Problem 4.26

Definitions and mathematical statement

For an integer $n\geq0$, let $\mathcal P_n$ consist of the complex polynomials $p(z)=1+a_1z+\cdots+a_nz^n$ satisfying $\operatorname{Re}p(z)>0$ whenever $|z|<1$. The degree may be smaller than $n$. Positivity is required in the open disk; the real part may vanish at boundary points.

Determine exactly \[ \Lambda_n=\max_{p\in\mathcal P_n} \int_0^{2\pi}|p(e^{i\theta})|^2\,d\theta =2\pi\max_{p\in\mathcal P_n} \left(1+\sum_{j=1}^{n}|a_j|^2\right). \] The equality on the right is Parseval's identity and fixes the normalization of the boundary integral. An extremal description should identify the maximizing coefficient vectors, allowing for rotations $p(z)\mapsto p(e^{it}z)$, which preserve both constraints and the objective.

The analytic constraint can equivalently be expressed by the nonnegativity, for all real $\theta$, of the trigonometric polynomial $1+\operatorname{Re}\sum_{j=1}^{n}a_je^{ij\theta}$. Indeed, boundary nonnegativity gives interior positivity because the value at the center is $1$. This formulation includes extremal polynomials with boundary zeros of the real part. Coefficient bounds for positive-real-part functions make the admissible coefficient set bounded; its boundary-nonnegative formulation makes it closed. Thus the maximum is an actual finite-dimensional maximum, not merely a supremum. The target is its sharp value and equality cases for every $n$.

Short English statement

Maximize the boundary square mean of a normalized polynomial with positive real part in the unit disk.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2304026 (AMR-022-4026), 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 4.26. Original source indexed by AMR-022-4026. https://arxiv.org/abs/1809.07200.