On maximal ranges of polynomial spaces in the unit disk |
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Authors: | Antonio Y Cordova Stephan Ruscheweyh |
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Institution: | 1. Universidad Técnica Federico Santa María, Valparaíso, Chile 2. Universit?t Würzburg, D-8700, Würzburg, FRG
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Abstract: | Let be a domain in C, 0, and let
n
0
() be the set of polynomials of degreen such thatP(0)=0 andP(D), whereD denotes the unit disk. The maximal range
n
is then defined to be the union of all setsP(D),P
n
0
(). We derive necessary and, in the case of ft convex, sufficient conditions for extremal polynomials, namely those boundaries whose ranges meet
n
. As an application we solve explicitly the cases where is a half-plane or a strip-domain. This also implies a number of new inequalities, for instance, for polynomials with positive real part inD. All essential extremal polynomials found so far in the convex cases are univalent inD. This leads to the formulation of a problem. It should be mentioned that the general theory developed in this paper also works for other than polynomial spaces.Communicated by J. Milne Anderson. |
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Keywords: | AMS classification" target="_blank">AMS classification 30C10 41A10 |
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