In this paper, we generalize several Berezin number inequalities involving product of operators, which acting on a Hilbert space . Among other inequalities, it is shown that if A,B are positive operators and X is any operator, then where , , , and .

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 ∈

In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are n × n $n\times n$ matrices, then ∥ A X B ∗ ∥ 2 ≤ ∥ f 1 ( A ∗ A ) X g 1 ( B ∗ B ) ∥ ∥ f 2 ( A ∗ A ) X g 2 ( B ∗ B ) ∥ , $$\begin{aligned} \bigl\Vert AXB^{*} \bigr\Vert ^{2}\leq \bigl\Vert f_{1} \bigl(A^{*}A\bigr)Xg_{1}\bigl(B^{*}B\bigr) \bigr\Vert \bigl\Vert f_{2}\bigl(A^{*}A\bigr)Xg_{2}\bigl(B^{*}B\bigr) \bigr\Vert , \end{aligned}$$ where f 1 $f_{1}$ , f 2 $f_{2}$ , g 1 $g_{1}$ , g 2 $g_{2}$ are non-negative continuous functions such that f 1 ( t ) f 2 ( t ) = t $f_{1}(t)f_{2}(t)=t$ and g 1 ( t ) g 2 ( t ) = t $g_{1}(t)g_{2}(t)=t$ ( t ≥ 0 $t\geq0$ ). We also obtain the inequality | | | A B ∗ | | | 2 ≤ | | | p ( A ∗ A ) m p + ( 1 − p ) ( B ∗ B ) s 1 − p | | | | | | ( 1 − p ) ( A ∗ A ) n 1 − p + p ( B ∗ B ) t p | | | , $$\begin{aligned} \bigl\vert \!\bigl\vert \!\bigl\vert AB^{*} \bigr\vert \!\bigr\vert \!\bigr\vert ^{2} &\leq \bigl\vert \!\bigl\vert \!\bigl\vert p \bigl(A^{*}A\bigr)^{\frac{m}{p}}+ (1-p) \bigl(B^{*}B\bigr)^{\frac {s}{1-p}} \bigr\vert \!\bigr\vert \!\bigr\vert \bigl\vert \!\bigl\vert \!\bigl\vert (1-p) \bigl(A^{*}A\bigr)^{\frac{n}{1-p}}+ p\bigl(B^{*}B \bigr)^{\frac{t}{p}} \bigr\vert \!\bigr\vert \!\bigr\vert , \end{aligned}$$ in which m, n, s, t are real numbers such that m + n = s + t = 1 $m+n=s+t=1$ , | | | ⋅ | | | $\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert $ is an arbitrary unitarily invariant norm and p ∈ [ 0 , 1 ] $p\in[0,1]$ .