Mathematics
http://dspaces.uok.edu.in/jspui//handle/1/1017
Thu, 25 Apr 2019 07:52:15 GMT2019-04-25T07:52:15ZBernstein-Type Inequalities for the Polar Derivative of a Polynomial
http://dspaces.uok.edu.in/jspui//handle/1/1204
Bernstein-Type Inequalities for the Polar Derivative of a Polynomial
Ahangar, Sajad Hussain (Scholar); Rather, Nisar Ahmad (Guide)
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.
Sun, 01 Jan 2012 00:00:00 GMThttp://dspaces.uok.edu.in/jspui//handle/1/12042012-01-01T00:00:00ZInequalities for the Derivative of a Polynomial
http://dspaces.uok.edu.in/jspui//handle/1/1203
Inequalities for the Derivative of a Polynomial
Dar, Bilal Ahmad (Scholar); Mir, M.A. (Guide)
Inequalities of Bernstein and Markov have been the starting point of a
considerable literature in approximation theory and our research shall be
centered on these inequalities. The problems of obtaining exact new bounds, the
improvements and generalizations of some older results for the maximum of
P'(z) over a unit disk are still of considerable interest. In view of this fact and
that of many as yet unsettled questions this subject continues in an active field of
research.
The aim of this dissertation is to present a survey of certain results
concerning the bounds for the maximum modulus of the derivative of an th n
degree polynomial over a disk with or without the restriction on the zeros of the
polynomial.
Sun, 01 Jan 2012 00:00:00 GMThttp://dspaces.uok.edu.in/jspui//handle/1/12032012-01-01T00:00:00ZBounds for the Moduli of the zeros of a Polynomial
http://dspaces.uok.edu.in/jspui//handle/1/1018
Bounds for the Moduli of the zeros of a Polynomial
Bhat, Faroz Ahmad (Scholar); Rather, Nisar Ahmad (Guide)
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.
Sat, 01 Jan 2011 00:00:00 GMThttp://dspaces.uok.edu.in/jspui//handle/1/10182011-01-01T00:00:00Z