If a polynomial $p(z)$ of degree $n$ does not assume the value $w$ for $|z|<1$ that is $p(z)\neq w$ for $|z|<1.$ Show that $p(z)-\dfrac{(1-e^{i\psi})}{n}zp^{\prime}(z)\neq w$ for $|z|<1,\psi\in\mathbb{R}.$

I know at least two solution of this problem one by using Luguerre's theorem concerning the polar derivative of a polynomial and another by using a known result. But I want to find its direct solution with using known results.