Microsoft Word - N.A. Rathar- _28-32_.doc N. A. Rather and Faroz Ahmad / BIBECHANA 9 (2013) 28-32 : BMHSS, p.28 (Online Publication: Nov., 2012) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA On the critical points of a polynomial N A Rather, Faroz Ahmad * Department of Mathematics, University of Kashmir * Corresponding author: Email: farozmaths1080@gmail.com Article history: Received 25 March, 2012; Accepted 13 August, 2012 Abstract Let denote the set of all polynomials of the form with and , . In this paper, we show that in for all polynomials . For , this reduces to a result due to Aziz and Zargar. Keywords: Polynomial, Critical point; Sendove conjecture; Walish Coincidence theorem. 1. Introduction The Gauss-Lucas Theorem states that if S is the set of zeros of a polynomial , then every zero of the derivative is contained in the smallest convex set that contains S. This is best possible, in the sense that, if has all its zeros in , then no proper subset of can be guaranteed to contain even one zero of , (as is shown by the polynomial of the form , since has zeros only at , which can lie anywhere in ). Gauss-Lucas theorem has been rather thoroughly investigated [8] and sharped in several ways. However, there is one related question that deserves attention, namely given one specific zero of , what can be said about a neighborhood of that will always contain a zero of . N. A. Rather and Faroz Ahmad / BIBECHANA 9 (2013) 28-32: BMHSS, p.29 (Online Publication: Nov., 2012) The following conjecture was made by Bulgarian Mathematician B L Sendov in 1962 but became later known as Ilief’s conjecture (See [4, problem 4.5] or [6, p.795]) . Conjecture 1. Let be a polynomial of degree having all its zeros in the unit disk . If is any one of these zeros, then has at least one zero in the disk . Since in 1962, when it is first became known, Conjecture 1 has been the subject of more than thirty articles. However, it was fully verified only for polynomials of degree (see [13]). A variety of special cases have been dealt with over the years (See [2, 7, 11] for references), among which we mention that of a polynomial with at most five distinct zeros [5], as well as Miller’s qualitative result [10], according to which those zeros of lying sufficiently close to the unit circle satisfy an even stronger condition that the one stated in Sendov’s conjecture (See also [12]). Another Stronger conjecture than that of Ilief was made in 1969 by Goodman, Rahman and Ratti [3]. Conjecture 2. Let be a polynomial of degree having all its zeros in the unit disk . If is any one of these zeros, then has at least one zero in the disk Conjecture 2 has been proved when [3], but some counter examples have been devised for case by M. J. Miller [10]. Recently Aziz and Zargar [2] have proved the following result . Theorem A. If is a polynomial of degree with , , then does not vanish in . In this paper we establish a generalized form of above theorem. In fact we prove the following interesting result which extracts that portion of complex plane in which the above polynomial does not vanish. Theorem 1. Let be a polynomial of degree with and , , then does not vanish in the disk N. A. Rather and Faroz Ahmad / BIBECHANA 9 (2013) 28-32: BMHSS, p.30 (Online Publication: Nov., 2012) . The result is best possible as shown by the polynomial , Further taking we get Theorem A. By using a similar argument, we can prove the following more general result. Theorem 2. Let be a polynomial of degree with and , where Then has fold zero at and remaining zeros of lie in the region . The result is best possible as shown by the polynomial , . For the proof of this theorem we need the following lemma which is the coincidence theorem of Walish [8, P.62] (see also [1]). Lemma. Let be a symmetric -linear form of total degree in and let C be a circular region containing the points , then there exists at least one point belonging to C such that . Proof of theorem 2. By hypothesis, where and Let , then is a polynomial of degree , having all its zeros in and we have . This implies . N. A. Rather and Faroz Ahmad / BIBECHANA 9 (2013) 28-32: BMHSS, p.31 (Online Publication: Nov., 2012) If now is any zero of , then from , we get (2) This is an equation which is linear and symmetric in the zeros of that is, in . Hence an application of the above lemma with circular region shows that will also satisfy the equation obtained by substituting into the equation (2) , where is suitably chosen point in the circular region . That is satisfies the equation or equivalently . Thus has the values or , where is suitably chosen point in . if , then using the fact that , it follows that If then clearly Thus in any case Since is an arbitrary zero of , it follows that every zero of lie in the disk This completes the proof of Theorem 2. Corollary . If we take in Theorem 2, we get Theorem 1. N. A. Rather and Faroz Ahmad / BIBECHANA 9 (2013) 28-32: BMHSS, p.32 (Online Publication: Nov., 2012) References [1] A. Aziz, Pacific J. Math., 118 (1985) 17-26. [2] Aziz and Zargar, Aus. Math. Soc., 57 (1998) 173-174. [3] A. W. Goodman, Q. I. Rahman and J. S. Ratti, Proc. Amer. Math . Soc., 21 (1969) 273-274. [4] W. K. Hayman, Research Problems in Function Theory, Athlone Press London (1967). [5] S. Kumar and B. G. Shenoy, J. Math. Anal. Appl., 171 (1992) 595-600. [6] M. Marden, Much ado about nothing, Amer. Math. Monthly, 83 (1976)788-789. [7] M. Marden, Amer. Math. Monthly, 90 (1983) 267-276. [8] M. Marden, Geometry of polynomials, IInd edition Math. Surveys, 3. Amer .Math. Soc., Providence, Rhode Island (1966). [9] A. Meir and A. Sharma, Pacific J. Math., 31 (1969) 459-467. [10] M. J. Miller, J. Math. Anal. Appl., 175 (1993) 632-639. [11] G. Schmeisser, Math. Z., 156 (1977) 165-173. [12] V.Vajaitu and Z. Zaharesen, London. Math. Soc., 25 (1993) 49-54. [13] Jonny E. Brown and Guangping Xiang, proof of the Sendov conjecture for polynomials of degree at most eight (1965) [14] Z. Rubinstien, Pacific J. Math., 26 (1968) 159-161.