Sharp Bernstein Inequalities for Jacobi–Dunkl Operators

We find sharp constants in the Bernstein inequality
$$ \|\Lambda_{\alpha,\beta}^rf\|\le M\|f\| $$
for the Jacobi–Dunkl differential-difference operator
$$ \Lambda_{\alpha,\beta}f(x) =f'(x)+\frac{A'_{\alpha,\beta}(x)}{A_{\alpha,\beta}(x)} \frac{f(x)-f(-x)}{2}\,. $$
Here $n,r\in\mathbb N$, $f$ is a trigonometric polynomial of degree $\le n$, the norm is uniform, $\alpha,\beta\ge -1/2$, and $A_{\alpha,\beta}(x)=(1-\cos x)^\alpha(1+\cos x)^\beta|{\sin x}|$ is the Jacobi weight. In the spaces $L_p$ with Jacobi weight, upper bounds are obtained.