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تسجيل مستخدم جديدInequalities for the derivative of Polynomials with restricted zeros
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In this paper we shall use the boundary Schwarz lemma of Osserman to obtain some generalizations and refinements of some well known results concerning the maximum modulus of the polynomials with restricted zeros due to Turan, Dubinin and others.
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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 $0 leq k leq n$.
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