Reverse Markov- and Bernstein-type inequalities for incomplete polynomials

Let ${mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$|f|_A := sup_{x in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A subset {Bbb R}$. Let $$V_a^b(f) := int_a^b{|f^{prime}(x)| , dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 frac nkleq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{|P(1)|}} leq c_2 left( frac nk + 1 right)$$ for all integers $n geq 1$ and $k geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac nkright)^{1/2} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{|P(1)|}} leq c_2 left(frac nk + 1right)^{1/2}$$ for all integers $n geq 1$ and $k geq 1$.