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We improve Mahler's lower bound for the Mahler measure in terms of thediscriminant and degree for a specific class of polynomials: complex monicpolynomials of degree $d\geq 2$ such that all roots with modulus greater thansome fixed value $r\geq1$ occur in equal modulus pairs. We improve Mahler'sexponent $\frac{1}{2d-2}$ on the discriminant to $\frac{1}{2d-3}$. Moreover, weshow that this value is sharp, even when restricting to minimal polynomials ofintegral generators of a fixed not totally real number field.  An immediate consequence of this new lower bound is an improved lower boundfor integral generators of number fields, generalising a simple observation ofRuppert from imaginary quadratic to totally complex number fields of arbitrarydegree.

On Mahler's inequality and small integral generators of totally complex number fields