  In this article, we present some new inequalities for the numerical radius of
products of Hilbert space operators and the generalized Aluthge transform. In
particular, we show some upper bounds for $\omega(ABC+DEF)$ using the
celebrated Buzano inequality, then some consequences that generalize some
results from the literature are discussed. After that, inequalities that
involve the generalized Aluthge transform are shown using some known bounds for
the numerical radius of the product of two operators.
