Inequalities for an operator on the space of polynomials

Let Pn be the class of all complex polynomials of degree at most n. Recently Rather et. al.[ On the zeros of certain composite polynomials and an operator preserving inequalities, Ramanujan J., 54(2021) 605–612. url{https://doi.org/10.1007/s11139-020-00261-2}] introduced an operator N:Pn→Pn defined by N[P](z):=∑kj=0λj(nz2)jP(j)(z)j!, k≤n where λj∈C, j=0,1,2,…,k are such that all the zeros of ϕ(z)=∑kj=0(nj)λjzj lie in the half plane |z|≤∣∣z−n2∣∣ and established certain sharp Bernstein-type polynomial inequalities. In this paper, we prove some more general results concerning the operator N:Pn→Pn preserving inequalities between polynomials. Our results not only contain several well known results as special cases but also yield certain new interesting results as special cases.