On the number of integral polynomialswith bounds placed on the derivative at $p$-adic and real roots

Consider a class of polynomials defined by a fixed degree and a fixed height. Introducing an additional constraint on the value of the $p$-adic norm of the derivative at a $p$-adic root, we find an upper bound on the number of such polynomials. A similar bound has been proved in the case where the derivative is bounded at a real and a $p$-adic root.