Exact $L_2$ Bernstein-Markov inequalities for generalized weights

In this paper, we obtain some exact $L_2$ Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem $$M_n^2(L_2(W_{\lambda }),\mathrm{D}):=\underset{0\mathrm{\ne }p\in {\mathcal{P}}_n}{\sup }\frac{{\int }_I{\mid \mathrm{D}p(x)\mid }^2{W}_{\lambda }(x)\mathrm{d}x}{{\int }_I\mathrm{\mid }p(x){\mathrm{\mid }}^2{W}_{\lambda }(x)\mathrm{d}x},\lambda >0,$$ where ${\mathcal{P}}_n$ denotes the set of all algebraic polynomials of degree at most $n$, $\mathrm{D}$ is the differential operator given by $$\mathrm{D}=\{\begin{array}{rlrl}{\displaystyle }& {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{o}\mathrm{r}{\mathcal{D}}_{\lambda },}& {\displaystyle }& {\displaystyle \mathrm{i}\mathrm{f}{W}_{\lambda }(x)=\mathrm{\mid }x{\mathrm{\mid }}^{2\lambda }{e}^{-x^2}\mathrm{a}\mathrm{n}\mathrm{d}I=\mathbb{R},}\\ {\displaystyle }& {\displaystyle (1-x^2{)}^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{o}\mathrm{r}(1-x^2{)}^{\frac{1}{2}}{\mathcal{D}}_{\lambda },}& {\displaystyle }& {\displaystyle \mathrm{i}\mathrm{f}{W}_{\lambda }(x):=\mathrm{\mid }x{\mathrm{\mid }}^{2\lambda }(1-x^2{)}^{\mu -\frac{1}{2}},\mu >-\frac{1}{2}\mathrm{a}\mathrm{n}\mathrm{d}I=[-1,1],}\end{array}$$ and ${\mathcal{D}}_{\lambda }$ is the univariate Dunkl operator, i.e., ${\mathcal{D}}_{\lambda }f(x)=f^{\mathrm{\prime }}(x)+\lambda (f(x)-f(-x))\mathrm{/}x$. Furthermore, the corresponding extremal polynomials are also obtained.