Some theorems on generalized polars with arbitrary weight
The present paper, which is a continuation of our earlier work in Annali di Mathematica [1] and Journal Math. Seminar [2] (EγEUθPIA), University of Athens, Greece, deals with the problem of determining sufficiency conditions for the nonvanishing of generalized polars (with a vanishing or nonvanishin...
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Online Access: | http://dx.doi.org/10.1155/S0161171287000851 |
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doaj-c11242df3c174b13b4f0f4d7cfab18562020-11-24T23:25:47ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251987-01-0110475777610.1155/S0161171287000851Some theorems on generalized polars with arbitrary weightNeyamat Zaheer0Aijaz A. Khan1Mathematics Department, King Saud University, Riyadh, Saudi ArabiaMathematics Department, Aligarh Muslim University, Aligarh, IndiaThe present paper, which is a continuation of our earlier work in Annali di Mathematica [1] and Journal Math. Seminar [2] (EγEUθPIA), University of Athens, Greece, deals with the problem of determining sufficiency conditions for the nonvanishing of generalized polars (with a vanishing or nonvanishing weight) of the product of abstract homogeneous polynomials in the general case when the factor polynomials have been preassigned independent locations for their respective null-sets. Our main theorems here fully answer this general problem and include in them, as special cases, all the results on the topic known to date and established by Khan, Marden and Zaheer (see Pacific J. Math. 74 (1978), 2, pp. 535-557, and the papers cited above). Besides, one of the main theorems leads to an improved version of Marden's general theorem on critical points of rational functions of the form f1f2…fp/fp+1…fq, fi being complex-valued polynomials of degree ni.http://dx.doi.org/10.1155/S0161171287000851generalized circular regionscircular conesgeneralized polarsabstract homogeneous polynomial. |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Neyamat Zaheer Aijaz A. Khan |
spellingShingle |
Neyamat Zaheer Aijaz A. Khan Some theorems on generalized polars with arbitrary weight International Journal of Mathematics and Mathematical Sciences generalized circular regions circular cones generalized polars abstract homogeneous polynomial. |
author_facet |
Neyamat Zaheer Aijaz A. Khan |
author_sort |
Neyamat Zaheer |
title |
Some theorems on generalized polars with arbitrary weight |
title_short |
Some theorems on generalized polars with arbitrary weight |
title_full |
Some theorems on generalized polars with arbitrary weight |
title_fullStr |
Some theorems on generalized polars with arbitrary weight |
title_full_unstemmed |
Some theorems on generalized polars with arbitrary weight |
title_sort |
some theorems on generalized polars with arbitrary weight |
publisher |
Hindawi Limited |
series |
International Journal of Mathematics and Mathematical Sciences |
issn |
0161-1712 1687-0425 |
publishDate |
1987-01-01 |
description |
The present paper, which is a continuation of our earlier work in Annali di
Mathematica [1] and Journal Math. Seminar [2] (EγEUθPIA), University of Athens,
Greece, deals with the problem of determining sufficiency conditions for the nonvanishing
of generalized polars (with a vanishing or nonvanishing weight) of the
product of abstract homogeneous polynomials in the general case when the factor
polynomials have been preassigned independent locations for their respective null-sets.
Our main theorems here fully answer this general problem and include in them,
as special cases, all the results on the topic known to date and established by Khan,
Marden and Zaheer (see Pacific J. Math. 74 (1978), 2, pp. 535-557, and the papers
cited above). Besides, one of the main theorems leads to an improved version of
Marden's general theorem on critical points of rational functions of the form
f1f2…fp/fp+1…fq, fi being complex-valued polynomials of degree ni. |
topic |
generalized circular regions circular cones generalized polars abstract homogeneous polynomial. |
url |
http://dx.doi.org/10.1155/S0161171287000851 |
work_keys_str_mv |
AT neyamatzaheer sometheoremsongeneralizedpolarswitharbitraryweight AT aijazakhan sometheoremsongeneralizedpolarswitharbitraryweight |
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1725556021769797632 |