Extremal problems for polynomials and their zero bounds

Abstract

newlinebounds is an outcome of the work carried out by author at the Post-Graduate newlineDepartment of Mathematics, University of Kashmir, Srinagar. This thesis consists newlineof four chapters all devoted to the polynomial inequalities, regions containing zeros newlineof the polynomials and zero free regions of analytic functions. The first chapter newlinecontains introduction and statement of results which are already present in the newlineliterature. newlineIn the second chapter, we studied the comparative position of the zeros of a newlinepolynomial which is derived by the composition of two polynomials and obtained newlinea beautiful result. As an application, we introduced a linear operator N : Pn and#8594; newlinePn, which preserves Bernstein type polynomial inequalities, hence established some newlinenew Bernstein-type polynomial inequalities for the operator N. Our results include newlineseveral well known polynomial inequalities as special cases. newlineIn the third chapter, by involving the moduli of the constant term and the newlineleading coefficient of a given polynomial, certain Turán-type inequalities for the newlinepolynomials are obtained. These estimates not only generalize the result of Dubinin newlinebut also strengthen some well known Turán-type inequalities for polynomials due to newlineMalik, Govil, Aziz, Aziz and Rather, Dubinin and others. newlineviIn the forth chapter, we have obtained results with relaxed hypothesis which newlinegive zero bounds for the larger class of polynomials a compared to those results newlineavailable in literature. In addition to this, we have also obtained some similar newlineresults for analytic functions. Our results not only generalize several well-known newlineresults but also provide better information abo......