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Register a new userOn an inequality concerning the polar derivative of a polynomial
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In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmunds inequality to the polar derivative of a polynomial.
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We present a new approach to the Marcinkiewicz interpolation inequality for the distribution function of the Hilbert transform, and prove an abstract version of this inequality. The approach uses logarithmic determinants and new estimates of canonical products of genus one.
In his 2006 paper, Jin proves that Kalantaris bounds on polynomial zeros, indexed by $m leq 2$ and called $L_m$ and $U_m$ respectively, become sharp as $mrightarrowinfty$. That is, given a degree $n$ polynomial $p(z)$ not vanishing at the origin and an error tolerance $epsilon > 0$, Jin proves that there exists an $m$ such that $frac{L_m}{rho_{min}} > 1-epsilon$, where $rho_{min} := min_{rho:p(rho) = 0} left|rhoright|$. In this paper we derive a formula that yields such an $m$, thereby constructively proving Jins theorem. In fact, we prove the stronger theorem that this convergence is uniform in a sense, its rate depending only on $n$ and a few other parameters. We also give experimental results that suggest an optimal m of (asymptotically) $Oleft(frac{1}{epsilon^d}right)$ for some $d ll 2$. A proof of these results would show that Jins method runs in $Oleft(frac{n}{epsilon^d}right)$ time, making it efficient for isolating polynomial zeros of high degree.
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The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with certain symmetry.
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