Applied Mathematics, 2011, 2, 1356-1358 doi:10.4236/am.2011.211189 Published Online November 2011 (http://www.SciRP.org/journal/am)
On the Zeros of a Polynomial Mohammad Syed Pukhta Division of Agricultural Engineering, Sher-e-Kashmir University of Agricultural Sciences & Technology of Kashmir, Srinagar, India E-mail:
[email protected] Received February 27, 2011; revised May 25, 2011; accepted June 3, 2011
Abstract In this paper we consider the problem of finding the estimate of maximum number of zeros in a prescribed region and the results which we obtain generalizes and improves upon some well known results. Keywords: Polynomial, Zeros, Complex Number, Prescribed Region an an 1 a1 a0 ,
1. Introduction n
Let p z ai z i be a polynomial of degree n such i 0
that
then the number of zeros of p z in z
1 does not 2
exceed n 1
an an 1 a1 a0 0
then according to a well known result of Enstrom and Kakeya, the polynomial p z , does not vanish in z 1. concerning the number of zeros of the polyno1 mial in the region z , the following result is due to 2 Mohammad [1]. n
Theorem A. Let p z ai z i be a polynomial of i 0
degree n such that 1 in z , does not 2
exceed 1
a 1 log n . ao log 2
Dewan [2] generalized Theorem A to the polynomials with complex coefficients and obtained the following result. n
Theorem B. If p z ai z i is a polynomial of dei 0
gree n with complex coefficients such that π arg ai , i 0,1, 2, , n for some real 2 and Copyright © 2011 SciRes.
i 0
a0
.
n
Theorem C. Let p z ai z i be a polynomial of i 0
degree n with complex coefficients. If Re ai i , Im ai i , for i 0,1, , n and n n 1 1 0 0, then the number of zeros
of p z in z
1 does not exceed 2 n
an an 1 a1 a0 0,
then the number of zeros of p z
1 log log 2
an cos sin 1 2sin ai
1
1 log log 2
n i i 0
a0
.
In this paper we generalize Theorem B and Theorem C under less restrictive conditions on the coefficients, which also improve upon them. More precisely, we prove the following. n
Theorem 1. Let p z ai z i be a polynomial of i 0
degree n with complex coefficients, such that π arg ai , i 0,1, 2, , n. for some real 2 and an an 1 a1 a0 0 then the number of zeros of p z in z , does not exceed AM
M. S. PUKHTA
If f z is regular, f 0 0 and f z M in z 1 , then ([4], p.171) the number of zeros of f z 1 M in z , 0 1 does not exceed . log 1 f 0 log
n 1 an cos sin 1 2sin ai 1 i 0 log 1 a 0 log
where 0 1. n
Theorem 2. Let p z ai z i be a polynomial of i 0
degree n with complex coefficients. If Re ai i , Im ai i , for i 0,1, , n and n n 1 1 0 0, n 0 , then the number of zeros of p z in z , 0 1 does not exceed n
1
1357
1 log
1
log
n i i 0
a0
.
Apply this result to g z in z does not exceed n 1 an cos sin 1 2sin ai 1 i 0 . log 1 a 0 log
All the number of zeros of p ( z ) in z is also equal to the number of zeros of g ( z ) in z . This completes proof of Theorem 1. Proof of Theorem 2. Consider n
g z 1 z p z an z n 1 ai ai 1 z i a0
2. Lemma
i 1
We need the following lemma for proof of the theorems. n
Lemma. Let p z ai z i be a polynomial of dei 0
gree n such that
The proof of the above lemma is omitted as it follows from the lemma in [3].
3. Proof of the Theorems Proof of Theorem 1. Consider the polynomial
g z 1 z p z 1 z an z n an 1 z n 1 a1 z a0 an z
n
g z an ai ai 1 a0 i 1
n
π arg ai ; ai ai 1 for some i 0,1, 2, , n, 2 then ai ai 1 ai ai 1 cos ai ai 1 sin .
n 1
For z 1 ,
an an 1 z
n
an 1 an 2 z n 1 a1 a0 z a0
For z 1 , we have n
g z an ai ai 1 a0 i 1
i 1 n
n
i 1
i 1
n n i i 1 i i 1 0 n 2 n i i 0 and using the same argument as in proof of Theorem 1, the proof of Theorem 2 follows. 1 Remark 1. For , Theorem 1 is a refinement of 2 1 Theorem B and for , and 0, it gives a 2 refinement of Theorem A. Remark 2. Theorem C can be deduced as a particular 1 case of Theorem 2 by putting . If we put 2 i 0, 0 i n in Theorem 2, we can deduce Theorem A. n
n
n
i 1
i 1
an ai ai 1 cos ai ai 1 sin a0
by using Lemma
n 1 an cos sin 1 2 ai sin i 0 a0 cos sin 1 n 1
an cos sin 1 2sin ai . i 0
Copyright © 2011 SciRes.
n n i i 1 i i 1 0 0
Corollary 1. Let p z ai z i be a polynomial of i 0
degree n, such that
n n 1 1 0 , then the number of zeros of p(z) in z , 0 1, does not exceed 1 1 log n . 1 a0 log
AM
M. S. PUKHTA
1358
4. Acknowledgements
[2]
Author is highly thankful to the referees for their valuable suggestions.
Q. G. Mohammad, “On the Zeros of the Polynomials,” American Mathematical Monthly, Vol. 72, No. 6, 1965, pp. 631-633. doi:10.2307/2313853
[3]
E. C. Titchmarsh, “The Theory of Functions,” 2nd Edition, Oxford University Press, London, 1939.
5. References
[4]
K. K. Dewan, “Extremal Properties and Coefficient Estimates for Polynomials with Restricted Zeros and on Location of Zeros of Polynomials,” Ph.D Thesis, Indian Institutes of Technology, Delhi, 1980.
[1]
N. K. Govil and Q. I. Rehman, “On the Enstrom Kakeya Theorem,” Tohoku Mathematical Journal, Vol. 20, No. 2, 1968, pp. 126-136. doi:10.2748/tmj/1178243172
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