1,820 research outputs found
Integral mean estimates for the polar derivative of a polynomial
Let be a polynomial of degree having all zeros in
where then it was proved by Dewan \textit{et al} that for every real
or complex number with and each
\indent In this paper, we shall present a refinement and generalization of
above result and also extend it to the class of polynomials
having
all its zeros in where and thereby obtain certain
generalizations of above and many other known results.Comment: 8 page
Maximal and inextensible polynomials and the geometry of the spectra of normal operators
We consider the set S(n,0) of monic complex polynomials of degree
having all their zeros in the closed unit disk and vanishing at 0. For we let denote the distance from the origin to the zero set of
. We determine all 0-maximal polynomials of degree , that is, all
polynomials such that for any .
Using a second order variational method we then show that although some of
these polynomials are inextensible, they are not necessarily locally maximal
for Sendov's conjecture. This invalidates the recently claimed proofs of the
conjectures of Sendov and Smale and shows that the method used in these proofs
can only lead to (already known) partial results. In the second part of the
paper we obtain a characterization of the critical points of a complex
polynomial by means of multivariate majorization relations. We also propose an
operator theoretical approach to Sendov's conjecture, which we formulate in
terms of the spectral variation of a normal operator and its compression to the
orthogonal complement of a trace vector. Using a theorem of Gauss-Lucas type
for normal operators, we relate the problem of locating the critical points of
complex polynomials to the more general problem of describing the relationships
between the spectra of normal matrices and the spectra of their principal
submatrices.Comment: A condensed version of the first half of this paper appeared in Math.
Scand., see arXiv:math/0601600. Parts of the second half appeared in Trans.
Amer. Math. Soc., see arXiv:math/0601519. The current version contains the
full details of the counterexample constructions and some other result
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