Nikolskii-type inequalities for algebra polynomials in regions with cusps

Let $G$ $\subset\mathbb{C}$ be a finite Jordan region, with $0\in G$, $L:=\partial G$; $\ P_{n}(z)$, $\deg P_{n}\leq n$, $n\in\mathbb{N}$, be an arbitrary algebraic polynomials and let $h(z)$ be a weight function. For $p>0$ we denote by $A_{p}(h,G)$ the class of analytic in $G$ functions $f$ such that
$$ \iint_{G}h(z)|f(z)|^{p}\,dx\,dy<\infty,\qquad z=x+iy; $$
and, when $L$ is rectifiable, by $\mathcal{L}_{p}(h,L)$, $p>0$, the class of measurable on $L$ functions $f$ such that
$$ \int_{L}h(z)|f(z)|^{p}\,|dz|<\infty. $$

In this work, we study the Nikol'skii-type inequalities for algebraic polynomials $P_{n}(z)$ and pointwise estimations for these polynomials in various regions of the complex plane through their $A_{p}(h,G)$ and $\mathcal{L}_{p}(h,L)$-norms, depending on the geometrical properties of regions and generalized Jacobi weight function $h(z)$ for some Jordan regions of complex plane.