Bounds for the Moduli of the zeros of a Polynomial

Polynomials pervade mathematics and much that is beautiful in mathematics is related to polynomials, virtually every branch of mathematics, from Algebraic number theory and Algebraic Geometry to Applied Analysis, Fourier analysis and Computer sciences, has its corpus of theory arising from the study of polynomials. Historically, give rise to some important problems of the day. The subject is now much too large to attempt an encyclopaedic coverage. The most complicated problems of trade and industry called for the solutions of equations and the introduction of literal symbols thus arose algebra, which at the time amounted to a science of equations. Even in antiquity, solutions had been for equations of first order and for quadratic equations, those stumbling blocks of today school children. We recall here that an expression of the form where are real or complex numbers with , is called a polynomial of degree . If there is a value of say, such that , then is called the zero of polynomial . Enormous efforts were put into solving polynomial equations of degree higher than the second and only in sixteenth century were such solutions forthcoming for equations of the third and fourth degrees. Another three centauries were spent in vain efforts to get the solutions of polynomial equations of degree higher than the fourth. It required the geneous of Abel and Galois to resolve this problem in it entirely. At the beginning of the nineteenth century, a young Norweigian mathematician, Neil Henrik Abel mediated long and Painstakingly on the problem and finally came to the conviction that equations of degree higher than fourth cannot, generally speaking, be solved by radicals. At about this time, another young mathematician Evarista Galois of France took a new approach and proved a similar result. The problems of obtaining exact new bounds, the improvements and generalisations of some older results for the location of the zeros of a polynomial are still of considerable interest. In view of this fact and that of many as yet unsettled questions this subject continues be an active field of research

Bernstein-Type Inequalities for the Polar Derivative of a Polynomial

Inequalities of Markov and Bernstein were the starting point of considerable literature in polynomial approximation theory. The problem of obtaining exact new bonds, the improvements and the extensions of some old results for the maximum modulus of P'(z) on the unit disk |z|=1 are still of considerable interest. In view of this fact and many unsettled problems, the analytic theory of polynomials continues to be an active field of research. The aim of this dissertation to present a survey of certain results concerning the estimates for the maximum modulus of the polar derivative of a polynomial on the unit disk with or without restriction on the zeroes of a polynomial