## Abstract

In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequality. In particular, we present

${w}_{p}^{p}({A}_{1}^{*}{T}_{1}{B}_{1},\mathrm{\dots},{A}_{n}^{*}{T}_{n}{B}_{n})\le \frac{{n}^{1-\frac{1}{r}}}{{2}^{\frac{1}{r}}}{\parallel \sum _{i=1}^{n}{[{B}_{i}^{*}{f}^{2}(|{T}_{i}|){B}_{i}]}^{rp}+{[{A}_{i}^{*}{g}^{2}(|{T}_{i}^{*}|){A}_{i}]}^{rp}\parallel}^{\frac{1}{r}}-\underset{\parallel x\parallel =1}{inf}\eta (x),$

where ${T}_{i},{A}_{i},{B}_{i}\in \mathbb{B}(\mathcal{\mathscr{H}})$
$(1\le i\le n)$, *f* and *g* are nonnegative continuous functions on $[0,\mathrm{\infty})$ satisfying $f(t)g(t)=t$ for all $t\in [0,\mathrm{\infty})$, $p,r\ge 1$, $N\in \mathbb{N}$, and

$\eta (x)={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^{n}}{\displaystyle \sum _{j=1}^{N}}(\sqrt[{2}^{j}]{{\u3008{({A}_{i}^{*}{g}^{2}(|{T}_{i}^{*}|){A}_{i})}^{p}x,x\u3009}^{{2}^{j-1}-{k}_{j}}{\u3008{({B}_{i}^{*}{f}^{2}(|{T}_{i}|){B}_{i})}^{p}x,x\u3009}^{{k}_{j}}}$$-\sqrt[{2}^{j}]{{\u3008{({B}_{i}^{*}{f}^{2}(|{T}_{i}|){B}_{i})}^{p}x,x\u3009}^{{k}_{j}+1}{\u3008{({A}_{i}^{*}{g}^{2}(|{T}_{i}^{*}|){A}_{i})}^{p}x,x\u3009}^{{2}^{j-1}-{k}_{j}-1}}){}^{2}.$

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