We study the sharp Markov--Bernstein--Nikol'skii inequality of the form

$\|D^{s}u\|_{\infty}\!\le C_{p}(n;s)\times\\\times\|u\|_{p}$, $p\in [1,\infty]$ for

trigonometric and algebraic polynomials $u$ of degree at most $n$ in the

weighted space $L^{p}$ with the Gegenbauer--Dunkl differential operator $D$. In

particular cases, these inequalities are reduced to the classical inequalities

of approximation theory of the Markov, Bernstein, and Nikol'skii type, to which

numerous papers are devoted. We apply the results of V.A. Ivanov (1983, 1992),

V.V. Arestov and M.V. Deikalova (2013, 2015), F. Dai, D.V. Gorbachev and

S.Yu. Tikhonov (2020) for algebraic constants in $L^{p}$ on compact Riemannian

manifolds of rank 1 (including the Euclidean sphere) and an interval with

Gegenbauer weight, refer to the works of E. Levin and D. Lubinsky (2015),

M.I. Ganzburg (2017, 2020), a review of the classic results of

G.V. Milovanovi\'c, D.S. Mitrinovi\'c and Th.M. Rassias (1994).

Earlier we studied the case $s=0$. In this paper, we consider the case $s\ge

0$. Our main result is to prove the existence in the trigonometric case for even $s=2r$ of

extremal polynomials $u_{*}$ that are real, even, and

$C(n;s)=\frac{|D^{s}u_{*}(0)|}{\|u_{*}\|_{p}}$. With the help of this fact, the

relationship with the algebraic constant for the Gegenbauer weight is proved.

On the one hand, this relationship allows to automatically characterize

extremal algebraic polynomials. On the other hand, well-known algebraic results

carry over to a more general trigonometric version. The main method of proof is

the application of the Gegenbauer--Dunkl harmonic analysis constructed by

D.V. Chertova (2009). As a consequence, we give the explicit constants for

$p=2,\,\infty$ (using the results of V.A. Ivanov), we give the relations of

orthogonality and duality (proved by methods of convex analysis from

approximation theory), we establish one asymptotic result of the

Levin--Lubinsky type (due to the connection with the multidimensional

Nikol'skii constant for spherical polynomials).