For every nonzero q with |q|<=1, the sharp spectral constant of the scaled q-numerical range is max{1, 2|q|/(1+sqrt(1-|q|^2))}.
A solution to Crouzeix's conjecture
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We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.
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Sharp spectral constants for scaled $q$-numerical ranges
For every nonzero q with |q|<=1, the sharp spectral constant of the scaled q-numerical range is max{1, 2|q|/(1+sqrt(1-|q|^2))}.