  Let $A = U |A|$ be the polar decomposition of $A$. The Aluthge transform of
the operator $A$, denoted by $\tilde{A}$, is defined as $\tilde{A}
=|A|^{\frac{1}{2}} U |A|^{\frac{1}{2}}$. In this paper, first we generalize the
definition of Aluthge transform for non-negative continuous functions $f, g$
such that $f(x)g(x)=x\,\,(x\geq0)$. Then, by using of this definition, we get
some numerical radius inequalities. Among other inequalities, it is shown that
if $A$ is bounded linear operator on a complex Hilbert space ${\mathscr H}$,
then
  \begin{equation*} h\left( w(A)\right) \leq \frac{1}{4}\left\Vert h\left(
g^{2}\left( \left\vert A\right\vert \right) \right) +h\left( f^{2}\left(
\left\vert A\right\vert \right) \right) \right\Vert +\frac{1}{2}h\left( w\left(
\tilde{A}_{f,g}\right) \right) , \end{equation*} where $f, g$ are non-negative
continuous functions such that $f(x)g(x)=x\,\,(x\geq 0)$, $h$ is a non-negative
non-decreasing convex function on $[0,\infty )$ and $\tilde{A}_{f,g} =f(|A|) U
g(|A|)$.
