 Applied Mathematics, 2011, 2, 1356-1358 doi:10.4236/am.2011.211189 Published Online November 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM On the Zeros of a Polynomial Mohammad Syed Pukhta Division of Agricultural Engineering, Sher-e-Kashmir University of Agricultural Sc iences & Technology of Kashmir, Srinagar, India E-mail: mspukhta_67@yahoo.co.in Received February 27, 2011; revised May 25, 2011; accepted J une 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 1. Introduction Let be a polynomial of degree n such that 0niiipz az1100nnaa aa then according to a well known result of Enstrom and Kakeya, the polynomial ,pz does not vanish in 1.z concerning the number of zeros of the polyno- mial in the region 1,2z the following result is due to Mohammad . Theorem A. Let be a polynomial of degree n such that 0niiipz az1100,nnaa aa then the number of zeros of pz in 1,2z does not exceed 11loglog 2noaa. Dewan  generalized Theorem A to the polynomials with complex coefficients and obtained the following result. Theorem B. If is a polynomial of de- 0niiipz azgree n with complex coefficients such that πarg,0,1, 2,,2iai11,nnaa aa0 then the number of zeros of in pz 12z does not exceed 100cossin12sin1loglog 2nniiaaa  . Theorem C. Let be a polynomial of 0niiipz azdegree n with complex coefficients. If Re, Im,ii iiaa for and 0,1, ,in1100,nn then the number of zeros of pz in 12z does not exceed 0011loglog 2nniia. 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. Theorem 1. Let be a polynomial of 0niiipz azdegree n with complex coefficients, such that πarg,0,1,2,,.2iain for some real  and 1100nnaa aa n for some real  and then the number of zeros of in pz ,z does not exceed M. S. PUKHTA1357 100cossin12sin1log1lognniaiaa  . where 01 Theorem 2. Let be a polynomial of 0niiipz azdegree n with complex coefficients. If Re, Im,ii iiaa for 0,1,,in0 and 11nn 0,0n, then the number of zeros of in pz ,0z1 does not exceed 0011log1lognniia. 2. Lemma We need the following lemma for proof of the theorems. Lemma. Let be a polynomial of de-gree n such that 0niiipz az1πarg ;2iiaaia for some 0,1,2,,,inthen 11 1cos siniii ii iaaa aa a  . The proof of the above lemma is omitted as it follows from the lemma in . 3. Proof of the Theorems Proof of Theorem 1. Consider the polynomial  1111111210011    nnnnnnnnnnnn0gzzpzzazaz azaazaazaaz aaza For 1z, we have  If fz is regular, and 00ffzM in 1, then (, p.171) the number of zeros of fz z101111110010cossin 0by using Lemmacossin1 2sincossin 1cossin1 2sin.nniiinnniiiiiinniinniigzaaaaaaa aaaaaaa     a in ,0 1z does not exceed 1log10logM. fApply this result to gz in z does not exceed 100cossin1 2sin1log1lognniiaaa  . All the number of zeros of in ()pz z is also equal to the number of zeros of ()gz in z. This completes proof of Theorem 1. Proof of Theorem 2. Consider 11011nniniiigzzpzaz aaza For 1z, 1011101111102nniiinn niiiiinnnniii iiinniigzaa aa00    and using the same argument as in proof of Theorem 1, the proof of Theorem 2 follows. Remark 1. For 1,2 Theorem 1 is a refinement of Theorem B and for 1,2 and 0, it gives a refinement of Theorem A. Remark 2. Theorem C can be deduced as a particular case of Theorem 2 by putting 12. If we put 0, 0iin in Theorem 2, we can deduce Theorem A. Corollary 1. Let be a polynomial of degree n, such that 0niiipz az11,nn 0 then the number of zeros of p(z) in ,01,z does not exceed 011log1logna. Copyright © 2011 SciRes. AM M. S. PUKHTA Copyright © 2011 SciRes. AM 1358 4. Acknowledgements Author is highly thankful to the referees for their valu-able suggestions. 5. References  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  Q. G. Mohammad, “On the Zeros of the Polynomials,” American Mathematical Monthly, Vol. 72, No. 6, 1965, pp. 631-633. doi:10.2307/2313853  E. C. Titchmarsh, “The Theory of Functions,” 2nd Edi-tion, Oxford University Press, London, 1939.  K. K. Dewan, “Extremal Properties and Coefficient Es-timates for Polynomials with Restricted Zeros and on Location of Zeros of Polynomials,” Ph.D Thesis, Indian Institutes of Technology, Delhi, 1980.