Further refinements of generalized numerical radius inequalities for Hilbert space operators

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 , … , A n * ⁢ T n ⁢ B n ) ≤ n 1 - 1 r 2 1 r ⁢ ∥ ∑ i = 1 n [ B i * ⁢ f 2 ⁢ ( | T i | ) ⁢ B i ] r ⁢ p + [ A i * ⁢ g 2 ⁢ ( | T i * | ) ⁢ A i ] r ⁢ p ∥ 1 r - inf ∥ x ∥ = 1 ⁡ η ⁢ ( x ) , w_{p}^{p}(A_{1}^{*}T_{1}B_{1},\dots,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1% }{r}}}{2^{\frac{1}{r}}}\bigg{\|}\sum_{i=1}^{n}[B_{i}^{*}f^{2}(|T_{i}|)B_{i}]^{% rp}+[A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i}]^{rp}\bigg{\|}^{\frac{1}{r}}-\inf_{\|x\|% =1}\eta(x), where T i , A i , B i ∈ ???? ⁢ ( ℋ ) {T_{i},A_{i},B_{i}\in\mathbb{B}(\mathscr{H})} ( 1 ≤ i ≤ n ) {(1\leq i\leq n)} , f and g are nonnegative continuous functions on [ 0 , ∞ ) {[0,\infty)} satisfying f ⁢ ( t ) ⁢ g ⁢ ( t ) = t {f(t)g(t)=t} for all t ∈ [ 0 , ∞ ) {t\in[0,\infty)} , p , r ≥ 1 {p,r\geq 1} , N ∈ ℕ {N\in\mathbb{N}} , and η ( x ) = 1 2 ∑ i = 1 n ∑ j = 1 N ( 〈 ( A i * ⁢ g 2 ⁢ ( | T i * | ) ⁢ A i ) p ⁢ x , x 〉 2 j - 1 - k j ⁢ 〈 ( B i * ⁢ f 2 ⁢ ( | T i | ) ⁢ B i ) p ⁢ x , x 〉 k j 2 j \displaystyle\eta(x)=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{N}\Bigl{(}\sqrt[2^{j% }]{\big{\langle}(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})^{p}x,x\big{\rangle}^{2^{j-1% }-k_{j}}\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i})^{p}x,x\big{\rangle}^{k_{j}}} - 〈 ( B i * ⁢ f 2 ⁢ ( | T i | ) ⁢ B i ) p ⁢ x , x 〉 k j + 1 ⁢ 〈 ( A i * ⁢ g 2 ⁢ ( | T i * | ) ⁢ A i ) p ⁢ x , x 〉 2 j - 1 - k j - 1 2 j ) 2 . \displaystyle -\sqrt[2^{j}]{\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i}% )^{p}x,x\big{\rangle}^{k_{j}+1}\big{\langle}(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})% ^{p}x,x\big{\rangle}^{2^{j-1}-k_{j}-1}}\,\Big{)}^{2}.