  In a recent work of the authors, we showed some general inequalities
governing numerical radius inequalities using convex functions. In this
article, we present results that complement the aforementioned inequalities. In
particular, the new versions can be looked at as refined and generalized forms
of some well known numerical radius inequalities. Among many other results, we
show that \[\left\| f\left( \frac{{{A}^{*}}A+A{{A}^{*}}}{4} \right) \right\|\le
\left\| \int_{0}^{1}{f\left( \left( 1-t \right){{B}^{2}}+t{{C}^{2}} \right)dt}
\right\|\le f\left( {{w}^{2}}\left( A \right) \right),\] when $A$ is a bounded
linear operator on a Hilbert space having the Cartesian decomposition $A=B+iC.$
This result, for example, extends and refines a celebrated result by kittaneh.
