The linear upper-bound conjecture for point evaluation on the circle

Let $\mathbb{T}$ be the unit circle, and let $L^p(\mathbb{T})$ have norm

$$\|f\|_p=\left(\int_{-\pi}^{\pi}|f(e^{i\theta})|^p\frac{d\theta}{2\pi}\right)^{1/p}.$$

For polynomials $P$ of degree at most $d$, define $\mathscr{C}_{d,p}$ to be the smallest constant $C$ such that

$$\|P\|_\infty^p\leq C\|P\|_p^p.$$

The linear upper-bound conjecture. If $d\geq 1$ and $p\geq 2$, then

$$\mathscr{C}_{d,p}\leq \frac{dp}{2}+1.$$

The conjecture improves the bound $\mathscr{C}_{d,p}\leq d\lceil p/2\rceil+1$ obtained from the power trick and Hölder's inequality. It is known for all $p\geq2$ when $d\leq4$, and for all $d$ when $p\geq6.8$; the remaining cases are open.