Some generalizations of the Aluthge transform of operators

Let $A=U\mathrm{\mid }A\mathrm{\mid }$ 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 $f(x)g(x)=x(x\ge 0)$. 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 $\mathcal{H}$, 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 $f(x)g(x)=x(x\ge 0)$, $h$ is a non-negative non-decreasing convex function on $[0,\mathrm{\infty })$ and $\tilde{A}{f,g} =f(|A|) U g(|A|)$.