A problem on relating the derivative of two associated polynomials

Question

If $P(z)=z-z_1$ and $Q(z)=z-(1/z_1)$ where $0<|z_1|<1,$ then $|z_1||P'(z)|=|z_1|<1=|Q'(z)|$ on $|z|=1.$ I am thinking of extending this property to polynomial of degree 2. If $P(z)=(z-z_1)(z-z_2)$ and $Q(z)=(z-(1/z_1))(z-(1/z_2)),$ where $0<|z_1|<1, 0<|z_1|\leq |z_2|<1,$ then I am trying to prove or disprove the following $|z_1|^2 \max_{|z|=1}|P'(z)| \leq\max_{|z|=1}|Q'(z)|.$ Kind request to share any thoughts on this claim.