# Turán-Type Reverse Markov Inequalities for Polynomials with Restricted Zeros Academic Article •
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### abstract

• 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}
ight)^{1/2} leq inf_{P}{frac{|P^{prime}|_{[-1,1]}}{|P|_{[-1,1]}}} leq c_2 left(frac{n}{k+1}
ight)^{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 Tur'an and Komarov from the case \$k=0\$ to the cases \$0 leq k leq n\$.

• Erdélyi, T.

### publication date

• January 1, 2020 11:11 AM