Some Inequalities on Polynomials in the Complex Plane Concerning a Linear Differential Operator

In this paper, we consider new extremal problems in the uniform norm between a univariate complex polynomial and its associated reciprocal polynomial involving a generalized B-operator. Our first result deals with inequality for the upper bound of a polynomial having s-fold zero at the origin governed by generalized B-operator, and as applications of this result, we are able to prove two inequalities for upper bounds of polynomials having s-fold zero at the origin considering the placement of the other zeros of the polynomials in ζ≥k,k≤1 under the condition that both comparing circles of radii r,R lie inside the circle of radius k or that one circle lies inside and the other outside. Further, we obtain an interesting upper bound for another class of polynomials having all its zeros in ζ≤k,k>0 under similar above conditions on the two comparing circles. Besides, a similar result for a self-inversive polynomial is also obtained. Moreover, each of the obtained results implicates some existing known estimates and their analogues for the extremum modulus of a polynomial.