Here you can find it for q-Bessel and for Bessel setting in references therein:

Article Radii of starlikeness and convexity of some q-Bessel functions

Thus, in principle, the rank one case of the Dunkl settings should (I would guess) follow from minor adaptations. For higher rank I have no idea of what can be done.

Thank you very much dear Prof. Luís Daniel Abreu for your response and your interest in my question.

It is interesting to see and think to the case in higher rank for Dunkl transform.

Now, I am interest in the rank one case of the Dunkl settings.

Using the convolution of harmonic functions, we introduce a generalization for a previously defined class of right half-strip harmonic mappings and determine sharp radii of univalence, full convexity and starlikeness for such functions