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Entire functions of exponential type not vanishing in the half-plane \(\Im z > k\), where \(k > 0\)

100%
Hachani M. A.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2017
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tom 71
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nr 1
EN
Let \(P (z)\) be a polynomial of degree \(n\) having no zeros in \(|z| < k\), \(k \leq 1\), and let \(Q (z) := z^n \overline{P (1/{\overline {z}})}\). It was shown by Govil that if \(\max_{|z| = 1} |P^\prime (z)|\) and \(\max_{|z| = 1} |Q^\prime (z)|\) are attained at the same point of the unit circle \(|z| = 1\), then \[\max_{|z| = 1} |P'(z)| \leq \frac{n}{1 + k^n} \max_{|z| = 1} |P(z)|.\]The main result of the present article is a generalization of Govil's polynomial inequality to a class of entire functions of exponential type.
2
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Some Algebraic Properties of Polynomial Rings

100%
Schwarzweller C., Korniłowicz A.
Formalized Mathematics
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2016
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tom 24
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nr 3
227-237
EN
In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.
3
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Factorization of weakly continuous holomorphic mappings

100%
González M., Gutiérrez J. M.
Studia Mathematica
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1996
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tom 118
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nr 2
117-133
EN
We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly continuous on weakly bounded sets if and only if it is weakly uniformly} continuous on weakly bounded sets. This result was obtained in 1983 by Aron, Hervés and Valdivia for polynomials between Banach spaces, and it also holds if the weak topology is replaced by a coarser one. However, we show that it need not be true for a stronger topology, thus answering a question raised by Aron. As an application of the first result, we prove that a holomorphic mapping ƒ between complex Banach spaces is weakly uniformly continuous on bounded subsets if and only if it admits a factorization of the form f = g∘S, where S is a compact operator and g a holomorphic mapping.
4
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Complementation in spaces of symmetric tensor products and polynomials

100%
Blasco F.
Studia Mathematica
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1997
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tom 123
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nr 2
165-173
EN
For a locally convex space E we prove that the space of n-symmetric tensors is complemented in the space of (n+1)-symmetric tensors endowed with the projective topology. Applications and related results are also given.
5
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Existence and determination of the set of Metzler matrices for given stable polynomials

88%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
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nr 2
389-399
EN
The problem of the existence and determination of the set of Metzler matrices for given stable polynomials is formulated and solved. Necessary and sufficient conditions are established for the existence of the set of Metzler matrices for given stable polynomials. A procedure for finding the set of Metzler matrices for given stable polynomials is proposed and illustrated with numerical examples.
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