scholarly journals The critical points of linear combinations of harmonic functions

1948 ◽  
Vol 54 (2) ◽  
pp. 196-206
Author(s):  
J. L. Walsh
Keyword(s):  
Critical Points ◽  
Harmonic Functions ◽  
Linear Combinations

The location of critical points of polynomials

2019 ◽  
Vol 12 (07) ◽  
pp. 1950087
Author(s):  
Suhail Gulzar ◽  
N. A. Rather ◽  
F. A. Bhat
Keyword(s):  
Complex Plane ◽  
Simple Proof ◽  
Critical Points ◽  
Classical Result ◽  
Zeros Of Polynomials ◽  
Linear Combinations ◽  
Incomplete Polynomials ◽  
Location Of Zeros ◽  
Set Of Points

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials.

scholarly journals A generalization of Lucas' theorem to vector spaces

1993 ◽  
Vol 16 (2) ◽  
pp. 267-276 ◽  
Author(s):  
Neyamat Zaheer
Keyword(s):  
Critical Points ◽  
Vector Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Linear Combinations ◽  
Inner Product Spaces ◽  
Complex Valued ◽  
Vector Valued

The classical Lucas' theorem on critical points of complex-valued polynomials has been generalized (cf. [1]) to vector-valued polynomials defined onK-inner product spaces. In the present paper, we obtain a generalization of Lucas' theorem to vector-valued abstract polynomials defined on vector spaces, in general, which includes the above result of the author [1] inK-inner product spaces. Our main theorem also deduces a well-known result due to Marden on linear combinations of polynomial and its derivative. At the end, we discuss some examples in support of certain claims.

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