International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 3, Issue 2 (September 2013) PP: 38-46 www.ijeijournal.com Page | 38 Zeros of a Polynomial in a given Circle M. H. Gulzar Department of Mathematics, University of Kashmir, Srinagar 190006 Abstract: In this paper we discuss the problem of finding the number of zeros of a polynomial in a given circle when the coefficients of the polynomial or their real or imaginary parts are restricted to certain conditions. Our results in this direction generalize some well- known results in the theory of the distribution of zeros of polynomials. Mathematics Subject Classification: 30C10, 30C15 Keywords and phrases: Coefficient, Polynomial, Zero. I. Introduction and Statement of Results In the literature many results have been proved on the number of zeros of a polynomial in a given circle. In this direction Q. G. Mohammad [5] has proved the following result: Theorem A: Let     0 ) ( j j j z a z P be a polynomial o f degree n such that 0 ...... 0 1 1       a a a a n n , Then the number of zeros of P(z) in 2 1  z does not exceed 0 log 2 log 1 1 a a n  . K. K. Dewan [2] generalized Theorem A to polynomials with complex coefficients and proved the following results: Theorem B: Let     0 ) ( j j j z a z P be a polynomial o f degree n such that j j a   ) Re( , j j a   ) Im( and 0 ...... 0 1 1           n n , Then the number of zeros of P(z) in 2 1  z does not exceed 0 0 log 2 log 1 1 a n j j n       . Theorem C: Let     0 ) ( j j j z a z P be a polynomial o f degree n with complex coefficients such that for some real   , , n j a j ,......, 2 , 1 , 0 , 2 arg        and 0 1 1 ...... a a a a n n      . Then the number f zeros of P(z) in 2 1  z does not exceed 0 1 0 sin 2 ) 1 sin (cos log 2 log 1 a a a n j j n          .