Turan-type reverse Markov inequalities for polynomials with restricted zeros

Let ${cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := {z in mathbb{C}: |z| leq 1, , , Im(z) geq 0}$$ be the closed upper half-disk of the complex plane. For integers $0 leq k leq n$ let ${mathcal F}_{n,k}^c$ be the set of all polynomials $P in {mathcal P}_n^c$ having at least $n-k$ zeros in $D^+$. Let $$|f|_A := sup_{z in A}{|f(z)|}$$ for complex-valued functions defined on $A subset {Bbb C}$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac{n}{k+1}right)^{1/2} leq inf_{P}{frac{|P^{prime}|_{[-1,1]}}{|P|_{[-1,1]}}} leq c_2 left(frac{n}{k+1}right)^{1/2}$$ for all integers $0 leq k leq n$, where the infimum is taken for all $0 otequiv P in {mathcal F}_{n,k}^c$ having at least one zero in $[-1,1]$. This is an essentially sharp reverse Markov-type inequality for the classes ${mathcal F}_{n,k}^c$ extending earlier results of Turan and Komarov from the case $k=0$ to the cases $0 leq k leq n$.