M.A.Kawoosa / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 3, 2014, pp 44-49
Zeros of Polynomial in Prescribed Region
M. A. Kawoosa
Dept. of Mathematics
Govt. P G , A . S .College 19000
Srinager , Kashmir , India
[email protected]
Abstract
The results concerning the number of zeros in a region has been of great use . In this paper we prove
some results concerning the number of zeros in prescribed region which improve and also generalize
some well known results with less restrictive conditions.
Keywords Polynomials, zeros, generalizations.
Mathematics Subject Classification(2000): 26C10, 12D10
1. INTRODUCTION:
The following results concerning the number of zeros in a closed disk is due to Q G Mohammad [9]
Theorem .A lf
is a polynomial of degree
Then the number of zeros of
in
,such that
does not exceed
Dewan[3] generalized Theorem A to the polynomials with complex coefficients and obtained the
following results.
Theorem. B lf
is a polynomial of degree with complex coefficients such that
then for some
and
then the number of zeros of
Theorem. C Let
and
in
does not exceed
be a polynomial of degree
,such that
where
are real numbers ,such that
44
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“ASRF promotes research nature, Research nature enriches the world’s future”
M.A.Kawoosa / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 3, 2014, pp 44-49
Then the number of zeros of
in
Theorem. D . lf
that
then for some
does not exceed
is a polynomial of degree
with complex coefficients such
and
in
then the number of zeros of
does not exceed
Where
In this paper we intend to generalize Theorem B, Theorem C and Theorem D with less restrictive
conditions on the coefficients and we prove the fallowing results.
2. RESULTS:
Theorem. 1 lf
that
is a polynomial of degree
then for some and
then the number of zeros of
in
with complex coefficients such
does not exceed
Where
To prove the theorem we need fallowing lemmas
Lemma.1 . Assume that
that
Let
is analytic in a disk
have zeros
, but not identically zero . Assume also
in
, then
The above lemma is well known Jensen ‘s Theorem []. The following lemma can be easily
deduced from lemma 1
Lemma.2. If
of
in
is regular ,
and
in
then the number of zeros
does not exceed
45
© IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved
“ASRF promotes research nature, Research nature enriches the world’s future”
M.A.Kawoosa / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 3, 2014, pp 44-49
Lemma. 3 lf
is a polynomial of degree
with complex coefficients such that
for some
Proof of theorem 1
Consider the polynomial
For
,we have
On using lemma 3 we get
46
© IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved
“ASRF promotes research nature, Research nature enriches the world’s future”
M.A.Kawoosa / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 3, 2014, pp 44-49
We now know from lemma 2 which states that , if
is regular ,
in
does not exceed
then the number of zeros of
in
On applying this to
which is regular and
does not exceed
and
, then the number of zeros of
Since all the zeros of
in
is also equal to the number of zeros of
therefore the Theorem 1 is proved completely.
Remark 1 if we put
generalizing Theorem D
in
in
the above Theorem1 reduces to Theorem D and hence
Remark 2 if we put
the above Theorem1 reduces to Theorem D and hence
generalizing Theorem B
The next Theorem gives the generalization of Theorem C
Theorem.2 Let
where
be a polynomial of degree
and
,such that
are real numbers ,such that
Then the number of zeros of
in
does not exceed
Proof of Theorem 2
Consider the polynomial
47
© IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved
“ASRF promotes research nature, Research nature enriches the world’s future”
M.A.Kawoosa / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 3, 2014, pp 44-49
Now for
, we have
We now know from lemma 2 which states that , if
is regular ,
in
does not exceed
then the number of zeros of
in
On applying this to
which is regular and
does not exceed
and
, then the number of zeros of
Since all the zeros of
in
is also equal to the number of zeros of
therefore the Theorem 2 is proved completely.
in
in
3.REFERENCES
[1].Lars V . Ahlfors, Complex Analysis, 3rd ed.,McGraw. Hill.
[2].H. Alzer, Bounds for the Zeros of polynomials Complex Variable, Theory Appl., 8(2001), 143-150.
[3].M. Bidkham and K.K . Dewan, On the zeros of a polynomial , Numerical Methods and Approximation Theory
111, (1987), 121-128.
[4].M.Dehmer, On the location of zeros of complex polynomials, Jipam, 4(2006), 1-13.
[5].N.K.Govil and Q. I. Rahman, On the Enestr m – Kakeya theorem , Tohoku Math. J., 20(1968), 126-136
[6].E. Kittaneh, Bounds for the zeros of polynomials from matrix inequalities , Arch. Math ., 81(2003), 601-608
[7].M. Marden , Geometry of polynomials, IInd ed. Mathematical Surveys , No 3 , Amer . math. Soc. Providence
R.I., 1966.
[8].G.V. Milovanovic , D. S. Mitrinovic and Th. M Rassias, Topics in polynomials, Extremal problems ,
Inequalities, Zeros , World Scientific Publishing Co., Singapore, 1994.
48
© IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved
“ASRF promotes research nature, Research nature enriches the world’s future”
M.A.Kawoosa / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 3, 2014, pp 44-49
[9].Q. G. Mohammad, , On the zeros of polynomials, Amer . Math. Monthly, 72 (1965) 631-633
[10].Q.I .Rahman and G. Schmeisser, Analytic theory of Polynomials, oxford University Press , New York,(2002)
[11].Ghulshan Singh and W.M.Shah,Number of Zeros of a Polynomial in a given Region, International Journal of
Modern Mathematical Sciences , 2012, 1(2):103-109
[12].A.Kawoosa and W.M.Shah, Bounds for the Zeros of a Polynomial with Perturbed Coefficients, International
Journal of Modern Mathematical Sciences, 2012, 3(1): 55-62.
[13].Mohammad syed Pukhta On the zeros of polynomial Applied mathematics ,2011,2,1356- 1358
AUTHOR’S BRIEF BIOGRAPHY
Prof Mujeeb Ahamad Kawoosa is a Senior Assistant Professor working
in higher education department presently posted at Govt A.S.College
,Srinagar Kashmir India. More the twelve paper has been published in
reputed international journals
49
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“ASRF promotes research nature, Research nature enriches the world’s future”