In the present article, we have attempted to reveal some relationships between a bounded linear operator $T$ acting on a Hilbert space and its generalized Aluthge transformation $T(s,t)$ in terms of their numerical ranges and norms. In fact, we have shown the following relations: \begin{enumerate} \item $\overline{W(f(T(t,1-t)))}\subseteq \overline{W(T)}$ for $t\in [0,1]$ and any rational function $f$. \item For an $n\times n$ matrix $T$, $T$ is convexoid iff $W(T)=W(T(t,1-t,))$ for all $t\in[0,1]$. \end{enumerate} \noindent (ii) is an extension of Ando's result in \cite{A_}.