B .A. Zargar and A . W. Manzoo / BIBECHANA 14 (2017) 48-53 : RCOST p.48 (Online Publication: Dec., 2016) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (Print), 2382-5340 (0nline) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Research Council of Science and Technology, Biratnagar, Nepal On zero-free regions for the derivative of a polynomial B .A. Zargar* and A . W. Manzoor Department of Mathematics, University of Kashmir Hazratbal Srinagar 190006, India *Email: bazargar@gmail.com Article history: Received 27 June, 2016; Accepted 17 August, 2016 DOI: http://dx.doi.org/10.3126/bibechana.v14i0.15521 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ Abstract Let Pn denote the set of all polynomials of the form      1 1 )( n j jzzzzp with .11,1  njz j In this paper we shall obtain some zero-free regions for the derivative of a polynomial. Keywords: Zero-free regions; Critical points; Sendov’s Conjecture. 1. Introduction Let us suppose that p(z) is an nth degree polynomial which has all its zeros in the unit disk 1z , then all the critical points of p(z) also lie in the same disk 1z .This is in fact the well- known Theorem which was implied in a note of Gauss dated 1836 and proved explicitly by Lucas dated 1874 (see also Marden [1]. Now instead of considering the relative position of all the zeros and critical points of p(z), let us choose any one zero zo of p(z) and ask: At most how far from zo does the nearest critical point lie ? A possible answer to this question is given by the following :Conjecture: “ If p(z) is an nth degree polynomial having all its zeros in the unit disk 1z and if zo is any one such zero, then at least one critical point of p(z) lie in the disk .10  zz ” This conjecture was included in the collection of Research Problems in Function Theory published in 1967 by professor Hayman [2], (see also [3]). Since it had been brought to Hayman’s attention by professor Ilyeff. It became B .A. Zargar and A . W. Manzoo / BIBECHANA 14 (2017) 48-53 : RCOST p.49 (Online Publication: Dec., 2016) known as “Ilyeff’s Conjecture”. Actually conjecture was due to a Bulgarian mathematician B. Sendov. In connection with this conjecture Brown [4] posed the following problem. Let nQ denote the set of all complex polynomials of the form      1 1 )( n j jzzzzp with .11,1  njz j Find the best constant Cn such that )(zp does not vanish in nCz  for all nQp . Brown observed that if 1)1()(  nzzzp then 0 1       n p and conjectured that . 1 n Cn  Recently Aziz and Zargar [5] settled this conjecture. Theorem1.1. Let      1 1 )( n k kzzzzp be a polynomial of degree n with ,11,1  nkzk then )(zp does not vanish in the disk . 1 n z  The result is best possible for the polynomial  20,)()( 1  niezzzp . First we shall prove the following interesting result which provides the zero free regions for the second derivative of polynomial      mn k j m zzzzp 1 )( Theorem 1.2. If      mn j j m zzzzp 1 )( where ,.,.........2,1,1 mnjz j  then the polynomial )(zp  does not vanish in . )1( )1( 0    nn mm z Taking m = 2 we get Corollary 1.If      2 1 2)( n j jzzzzp where ,2.,.........2,1,1  njz j then the polynomial )(zp  does not vanish in . )1( 2 0   nn z It is clearly of interest to known that a zero free region for the polynomial )(zpm where      mn j j m zzzzp 1 )( In this direction, we prove the following interesting results: B .A. Zargar and A . W. Manzoo / BIBECHANA 14 (2017) 48-53 : RCOST p.50 (Online Publication: Dec., 2016) Theorem1.3. Let      mn j j m zzzzp 1 )( be a polynomial of degree n, with ,.,.........2,1,1 mnjz j  then the polynomial )(zpm does not vanish in the disk . )1)....(1( !   mnnn m z Remark 1. If m=1,then we get Theorem 1.1. For the proofs of these theorems we need the following result which is due to Aziz and Zagar [5]. Lemma: Let      mn j j m zzzzp 1 )( where ,1,1 mnjz j  then )(zp does not vanish in .0 n m z  2. Proofs of Theorems Proof of Theorem 1.2. We write, )()( zQzzp m where   .,......2,1,1,)( 1 mnjzzzzQ j mn j j    By above lemma, the polynomial )()()( 1 zQmzzQzzp mm  )(1 zRzm , where ),()()( zmQzQzzR  does not vanish in .0 n m z  Replacing zby ,z n m it follows that the polynomial,       z n m pzS )(               z n m Rz n m m m 1 1 does not vanish in 10  z ,so that all the zeros of       z n m R lie in .1z B .A. Zargar and A . W. Manzoo / BIBECHANA 14 (2017) 48-53 : RCOST p.51 (Online Publication: Dec., 2016) Using the above lemma again and noting that S(z) is a polynomial of degree n-1, it follows that )(zS  does not vanish in . 1 1 0    n m z Or equivalently,       z n m p does not vanish in 1 1 0    n m z Replacing z by ,z n m it follows that     )(1)( 21 zRzmzRzzp mm    )()1()(2 zRmzRzzm   )(2 zTzm where     )(1)( zzRmzRzzT  does not vanish in 1 1 0    n m z . This completes the proof of Theorem 1.2. Proof of Theorem 1.3. By hypothesis, )()( zQzzp m where   mnjzzzzQ j mn j j    ,......2,1,1,)( 1 . By the above lemma, the polynomial )(zp does not vanish in .0 n m z  Therefore Theorem 1.2 yeilds that   )(2 zTzzp n , where     )(1)( zzRmzRzzT  does not vanish in . )1( )1( 0    nn mm z Replacing z by , it follows that z nn mm )1( )1(   B .A. Zargar and A . W. Manzoo / BIBECHANA 14 (2017) 48-53 : RCOST p.52 (Online Publication: Dec., 2016)         z nn mm pzU )1( )1( )(                   z nn mm Tz nn mm n )1( )1( )1( )1( 2 does not vanish in ,10  z so that all the zeros of         z nn mm T )1( )1( lie in 1z .Applying the above lemma again and noting that U(z) is a polynomial of degree n-2, thus it implies that )(zU  does not vanish in . 2 2 0    n m z Or equivalently,         z nn mm p )1( )1( does not vanish in . 2 2 0    n m z Replacing z by z mm nn )1( )1(   it follows that, )()2()()( 32 zTznzTzzp nn    )()2()(3 zTnzTzz n   )(3 zVz n , where )()2()()( zTnzTzzV  does not vanish in . )2)(1( )2)(1( 0    nnn mmm z In a similar way we see that the polynomial )()3()3()()( 43 zVznnzVzzp nniv   )(4 zWz n where, )()3()()( zVnzVzzW  does not vanish in . )3)(2)(1( )3)(2)(1( 0    nnnn mmmm z Proceeding in this way and noting that m and n are positive integers it follows that the polynomial does not vanish in )1).....(1( ! )1)....(1( 1.2).......1(      mnnn m mnnn mm z Which proves Theorem 1.3. B .A. Zargar and A . W. Manzoo / BIBECHANA 14 (2017) 48-53 : RCOST p.53 (Online Publication: Dec., 2016) References [1] M. Marden, Geometry of polynomials, Math. Survey’s No.3, American Math.Soc. (1966). [2] W. K. Hayman, Research Problems in Function Theory, London (1967) pp. 1-25. [3] Q. I. Rahman and G. Schmesser’s Analytic Theory of polynomials, Oxford University Press,New York (2002). [4] J. E. Brown, On theIlief-Sendov Conjecture, Pacific J.Math.135 (1988) 223-232. http://dx.doi.org/10.2140/pjm.1988.135.223 [5] A. Aziz and B.A. Zargar, On the critical points of a polynomial, Bull. Austral. Math. Soc.57 (1998) 173-174. http://dx.doi.org/10.1017/S000497270003152X