• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.45-72
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• 2022

In this work, we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with piecewise smooth boundary without cusps of the complex plane. Also, estimates are given on the whole complex plane.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.43-55
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• 1997

Suppose that D is a domain in the extended complex plane $\overline{C} = C \cup {\infty}$. For each $z_0 \in C$ and $C < r < \infty$, we let $B(z_0, r) = {z \in C : $\mid$z - z_0$\mid$ < r}$ and $S(z_0, r) = \partial B(z_0, r)$. For non-empty sets A, $B \subset \overling{C}$, diam (A) is the diameter of A and d(A, B) is the distance of A and B.

• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.151-170
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• 2009

Let m be a positive integer. We show that for any given real number ${\alpha}\;{\in}\;[0,\;1]$ and complex number $\mu$ with $|\mu|{\leq}1$ which satisfy $e^{2{\pi}i{\alpha}}{\mu}^m\;{\neq}\;1$, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number $\alpha$. Moreover if the given real number $\alpha$ is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.