  Let $\mathscr{P}_n $ denote the space of all complex polynomials
$P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$ and $\mathcal{B}_n$ a family of
operators that maps $\mathscr{P}_n$ into itself. In this paper, we consider a
problem of investigating the dependence of $$|B[P\circ\sigma](z)-\alpha
B[P\circ\rho](z)+\beta\{(\frac{R+k}{k+r})^{n}-|\alpha|\}B[P\circ\rho](z)| $$ on
the maximum and minimum modulus of $|P(z)|$ on $|z|=k$ for arbitrary real or
complex numbers $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq
1,R>r\geq k,$ $\sigma(z)=Rz,$ $\rho(z)=rz$ and establish certain sharp operator
preserving inequalities between polynomials, from which a variety of
interesting results follow as special cases.
