In this paper, we prove that for
$\ell \,=\,1$ or 2 the rate of best
$\ell $ - monotone polynomial approximation in the
${{L}_{p}}$ norm
$\left( 1\,\le \,p\,\le \,\infty\right)$ weighted by the Jacobi weight
${{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}$ with
$\alpha ,\,\beta \,>\,-1/p$ if
$p\,<\,\infty $ , or
$\alpha ,\,\beta \,\ge \,0$ if
$p\,=\,\infty $ , is bounded by an appropriate
$\left( \ell \,+\,1 \right)$ -st modulus of smoothness with the same weight, and that this rate cannot be bounded by the
$\left( \ell \,+\,2 \right)$ -nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.