  In this work, a pre-Gr\"{u}ss inequality for positive Hilbert space operators
is proved. So that, some numerical radius inequalities are proved. On the other
hand, based on a non-commutative Binomial formula, a non-commutative upper
bound for the numerical radius of the summand of two bounded linear Hilbert
space operators is proved. A commutative version is also obtained as well.

Functional Analysis