1. Investigation of the CrouzeixPalencia proof
This section deals with the careful analysis of the proof in order to think of possible adaption in order to improve the constant.
Problem 1.1.
[E. Wegert] Better estimate for $fg$ than $\fg\\leq \f\\g\$. Modifications of the Cauchy integral, especially concerning its behavior on constant functions.
Remark. see A. Greenbaum’s talk


Problem 1.2.
[T. Ransford] Let $\theta$ be a continuous homomorphism from a uniform algebra $A$ into a $C^{*}$algebra $B$ and $\alpha:A\rightarrow A$ be antilinear and such that [*] $\\alpha f\\leq \f\$ for all $f\in A$,
 [*] $\\theta f +(\theta\alpha f)^{*}\\leq 2\f\$ for all $f\in A$, and
 [*] $\alpha 1=1$.

Problem 1.3.
[J. Burke] Does it help to replace $\lambda^2$ in the proof by $p(\lambda)$ for polynomial other than $\lambda^2$? 
Problem 1.4.
[T. Ransford] Let $f$ be such that $f(z)=1$ for $z\in\partial\Omega$. Then $$C(\bar{f},z)=\frac{1}{f(z)}\sum_{\xi_{j}=\text{zero of }f} \frac{R(\frac{1}{f},\xi_{j})}{z\xi_{j}}.$$ How to prove that $C(\bar{f},z)\leq 1$? (what about Blaschke products?) 
Problem 1.5.
[T. Ransford] Let $f$ be such that $f(z)=1$ for $z\in\partial\Omega$. Then $$C(\bar{f},z)=\frac{1}{f(z)}\sum_{\xi_{j}=\text{zero of }f} \frac{R(\frac{1}{f},\xi_{j})}{z\xi_{j}}.$$ How to prove that $C(\bar{f},z)\leq 1$ directly?
Remark. B. Beckermann: For $z\in\partial \Omega$, $C(\bar{f},z)=$ Kovari et al. Green’s formula “vanished” in the recent developments, Faber operators


Problem 1.6.
[A. Greenbaum] Can anything be said about $\delta,\hat{\delta}$, in particular $$\delta_{\phi(A)}=\int_{\partial \mathbb{D}}\lambda_{min}(\mu(\sigma(s),\phi(A))ds?$$ 
Problem 1.7.
[A. Greenbaum] In what situations (other than in CaldwellGreenbaumLi) does the estimate $$\f(A)+g(A)^{*}\\geq \f(A)\$$ hold? Here, $f=B\circ \phi$ is such that $B$ is a Blaschke product, $\phi$ is a conformal mapping from $\Omega$ to $\mathbb{D}$ and such that $\f(A)\$ is maximized over all all $\f\_{\Omega}=1$. 
Problem 1.8.
[A. Greenbaum] What if $\Omega$ consists of a union of disjoint regions? It still holds that $$\f(A)+g(A)^{*}\\leq 2 +\delta,$$ but how does is this related to $\f(A)\$? What is the optimal $f$, like in the case of the union of two disks?
Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.