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Register a new userInequalities 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 $0 leq k leq n$.
In a previous work, we introduced the Collatz polynomials; these are the polynomials $left[P_N(z)right]_{Ninmathbb{N}}$ such that $left[z^0right]P_N = N$ and $left[z^{k+1}right]P_N = cleft(left[z^kright]P_Nright)$, where $c:mathbb{N}rightarrow mathbb{N}$ is the Collatz function $1rightarrow 0$, $2nrightarrow n$, $2n+1rightarrow 3n+2$ (for example, $P_5(z) = 5 + 8z + 4z^2 + 2z^3 + z^4$). In this article, we prove that all zeros of $P_N$ (which we call Collatz zeros) lie in an annulus centered at the origin, with outer radius 2 and inner radius a function of the largest odd iterate of $N$. Moreover, using an extension of the Enestrom-Kakeya Theorem, we prove that $|z| = 2$ for a root of $P_N$ if and only if the Collatz trajectory of $N$ has a certain form; as a corollary, the set of $N$ for which our upper bound is an equality is sparse in $mathbb{N}$. Inspired by these results, we close with some questions for further study.
We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence $p_{k+1}=x p_k-c_k p_{k-1},$ with a nondecreasing sequence ${c_k}$. As a special case they include a non-asymptotic version of Mate, Nevai and Totik result on the largest zeros of orthogonal polynomials with $c_k=k^{delta} (1+ o(k^{-2/3})).$
We introduce and analyze some numerical results obtained by the authors experimentally. These experiments are related to the well known problem about the distribution of the zeros of Hermite--Pade polynomials for a collection of three functions $[f_0 equiv 1,f_1,f_2]$. The numerical results refer to two cases: a pair of functions $f_1,f_2$ forms an Angelesco system and a pair of functions $f_1=f,f_2=f^2$ forms a (generalized) Nikishin system. The authors hope that the obtained numerical results will set up a new conjectures about the limiting distribution of the zeros of Hermite--Pade polynomials.
A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.
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