Roots of convex combination of two polynomials with real coefficient

Question

Suppose we have two monic polynomials $p_1(x), p_2(x)$ with real coefficients of the same degree. If the roots of both polynomials are in the disk $D = \{ z \in \mathbb C : |z| < R\}$. Let $p_t(x) = (1-t) p_1(x) + t p_2(x)$ where $t \in [0,1]$. What can we say about the roots of $p_t(x)$? Will the roots be in the disk $D$?