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A generalization of the Gauss-Lucas theorem

J. L. Díaz-Barrero, J. J. Egozcue (2008)

Czechoslovak Mathematical Journal

Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

Let p(z) be a polynomial of the form p ( z ) = j = 0 n a j z j , a j - 1 , 1 . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix

Fuad Kittaneh (2003)

Studia Mathematica

It is shown that if A is a bounded linear operator on a complex Hilbert space, then w ( A ) 1 / 2 ( | | A | | + | | A ² | | 1 / 2 ) , where w(A) and ||A|| are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.

A pure smoothness condition for Radó’s theorem for α -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

Let Ω n be a bounded, simply connected -convex domain. Let α + n and let f be a function on Ω which is separately C 2 α j - 1 -smooth with respect to z j (by which we mean jointly C 2 α j - 1 -smooth with respect to Re z j , Im z j ). If f is α -analytic on Ω f - 1 ( 0 ) , then f is α -analytic on Ω . The result is well-known for the case α i = 1 , 1 i n , even when f a priori is only known to be continuous.

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