  In this paper, we show some refinements of generalized numerical radius
inequalities involving the Young and Heinz inequalities. In particular, we
present
  \begin{align*}
w_{p}^{p}(A_{1}^{*}T_{1}B_{1},...,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1}{r}}}{2^{\frac{1}{r}}}\Big\|\sum_{i=1}^{n}[B_{i}^{*}
f^{2}(|T_{i}|)B_{i}]^{rp}+[A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i}]^{rp}\Big\|^{\frac{1}{r}}
-\inf_{\|x\|=1}\eta(x), \end{align*} where $T_{i}, A_{i}, B_{i} \in {\mathbb
B}({\mathscr H})\,\,(1\leq i\leq n)$, $f$ and $g$ are nonnegative continuous
functions on $[0, \infty)$ satisfying $f(t)g(t)=t$ for all $t\in [0, \infty)$,
$p, r\geq 1$, $N\in {\mathbb N}$ and \begin{align*} \eta(x)=
\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{N} \Big(\sqrt[2^{j}]{ \langle
(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})^{p}x, x\rangle^{2^{j-1}-k_{j}} \langle
(B_{i}^{*} f^{2}(|T_{i}|)B_{i})^{p}x, x\rangle^{k_j}}\quad-\sqrt[2^{j}]{
\langle (B_{i}^{*}f^{2}(|T_{i}|)B_{i})^{p}x, x\rangle^{k_{j}+1} \langle
(A_{i}^{*} g^{2}(|T_{i}^{*}|)A_{i})^{p}x, x\rangle^{2^{j-1}-k_{j}-1}}\Big)^{2}.
  \end{align*}
