On the norms of generalized translation operators generated by Jacobi–Dunkl operators

We establish an integral representation and improve the norm estimate for the generalized translation operators generated by Jacobi–Dunkl operators
$$ \Lambda_{\alpha,\beta}f(x)=f'(x)+\frac{A_{\alpha,\beta}'(x)}{A_{\alpha,\beta}(x)}\,\frac{f(x)-f(-x)}2, $$
where
$$ A_{\alpha,\beta}(x)=(1-\cos x)^\alpha(1+\cos x)^\beta|\sin x|, $$
in the spaces $L_p[-\pi,\pi]$ with the weight $A_{\alpha,\beta}$. For $\alpha\ge\beta\ge-\frac12$ we prove that these norms do not exceed $2$.