On the zeros of a class of generalized derivatives
- PDF / 1,147,213 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 19 Downloads / 250 Views
On the zeros of a class of generalized derivatives N. A. Rather1 · A. Iqbal1 · Ishfaq Dar1 Received: 17 December 2019 / Accepted: 11 August 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract In this paper, we obtain some results concerning the zeros of a class of generalized derivatives which are analogous to those for the ordinary derivative and the polar derivatives of polynomials. Keywords Polynomials · Zeros · Critical points · Convex sets · Polar derivatives Mathematics Subject Classification 26D10 · 41A17 · 30C15
1 Introduction For each positive integer n, let ℙn denote the linear space of all polynomials of degree at most n over the field ℂ of complex numbers, 𝜕ℙn denote the collection of all monic polynomials of degree n in ℙn and ℝn+ be the set of all n-tuples 𝛾 = (𝛾1 , 𝛾2 , … , 𝛾n ) of positive real ̂ be the extended complex plane (Riemann numbers with 𝛾1 + 𝛾2 + ⋯ + 𝛾n = n . Further, if ℂ ̂ , we mean either a closed disk, or a Sphere), then by a circular region [2, see p. 48] in ℂ closed half plane, or the complement of an open disk. Recall for f ∈ ℙn , all points z where f(z) vanishes are called the zeros of f and all points z where f � (z) vanishes are called the critical points of f. In other words the critical points of f are simply the zeros of its first derivative. The relationship between the zeros and the critical points of a polynomial is given by the following theorem, [3, See p. 72].
Theorem A Every critical point of a polynomial f can be expressed as a convex linear combination of its zeros.
An immediate consequence of Theorem A is the following
* A. Iqbal [email protected] N. A. Rather [email protected] Ishfaq Dar [email protected] 1
Department of Mathematics, University of Kashmir, Srinagar 190006, India
13
Vol.:(0123456789)
N. A. Rather et al.
Theorem B (Gauss–Lucas) Every convex set containing all the zeros of a polynomial also contains all its critical points. Theorem B guarantees that a closed disk containing all the zeros of a polynomial also contains all its critical points. Since we may map a circular region onto a closed disk via a linear fractional transformation
z = 𝜓(w) =
aw + b , cw + d
a, b, c, d ∈ ℂ,
with
ad ≠ bc,
one might expect that any circular region C containing all the zeros of a polynomial also contains all its critical points. This is certainly true if C is either a closed disk or a closed half plane, but need not be true when C is the complement of an open disk, and the same ̂ ∶ |z| ≥ 1} . To understand why this can be seen by taking f (z) = z3 − 8 with C = {z ∈ ℂ happens, let f (z) =( a0 +)az + a2 z2 + ⋯ + an−1 zn−1 + an zn be a polynomial of degree n and g(w) = (cw + d)n f aw+b be its transform under 𝜓 . In the case c = 0 , g is also a polynomial cw+d of degree n with its zeros and critical points related to those of f in the following natural fashion • w is a zero of g implies 𝜓(w) is a zero of f, • w⋆ is a critical point of g implies 𝜓(w⋆ ) is a critical point of f.
That is, when c = 0, the zeros and
Data Loading...
