Abdul Liman

TL;DR: In this paper, the authors consider an operator Dα which maps a polynomial P(z) into DαP(z), and prove results concerning the estimates of |DαP| on the disk |z| = R ≥ 1.

Abstract: In this paper we consider an operator B which carries a polynomial P(z) of degree n into B[P(z)]= λ0P(z) + λ1(nz/2)P’(z)/1! + λ2 (nz/2)2P”(z)/2! Where λ0, λ1 and λ2 are such that all the zeros of U(z)= λ0 + C(n, 1)λ1z + C(n, 2) λ2 z2 lie in the half plane |z|≤|z-n/2| and investigate the dependence of |B[P(Rz)] – α B[P(rz)]| on the minimum and the maximum modulus of P(z) on for every real or complex number α with |α|≤ 1 , R > r ≥ 1 with restriction on the zeros of the polynomial P(z) and establish some new operator preserving inequalities between polynomials.

TL;DR: In this article, the authors considered a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree n and proved some sharp results concerning Bernstein type inequalities for rational functions.

TL;DR: A generalization and an improvement of the polynomial inequalities of similar nature for R ⩾ 1 and ∣ z ∣–⩽ 1.