  In this work, an improvement of H\"{o}lder-McCarty inequality is established.
Based on that, several refinements of the generalized mixed Schwarz inequality
are obtained. Consequently, some new numerical radius inequalities are proved.
New inequalities for numerical radius of $n\times n$ matrix of Hilbert space
operators are proved as well. Some refinements of some earlier results were
proved in literature are also given. Some of the presented results are refined
and it shown to be better than earlier results were proved in literature.
