We propose some refinements for the second inequality in \( \frac{1}{2}\left\Vert A\right\Vert \le w(A)\le \left\Vert A\right\Vert, \) where A ∈ B(H). In particular, if A is hyponormal, then, by refining the Young inequality with the Kantorovich constant K K(⋅, ⋅), we show that \( w(A)\le \frac{1}{2{\operatorname{inf}}_{\left\Vert x\right\Vert =1}\zeta (x)}\left|\left\Vert A\right.\right|++\left|\left.{A}^{\ast}\right\Vert \right|\le \frac{1}{2}\left|\left\Vert A\right.\right|+\left|\left.{A}^{\ast}\right\Vert \right|, \) where \( \upzeta (x)=K{\left(\frac{\left\langle \left|A\right|x,x\right\rangle }{\left\langle \left|{A}^{\ast}\right|x,x\right\rangle },2\right)}^r,r=\min \left\{\uplambda, 1-\uplambda \right\}, \) and 0 ≤ λ ≤ 1. We also give a reverse for the classical numerical radius power inequality w(Aⁿ) ≤ wⁿ(A) for any operator A ∈ B(H) case where n = 2.

我们建议在第二个不等式的一些改进\（\压裂{1} {2} \左\韦尔A \右\韦尔\乐W（A）\乐\左\韦尔A \右\韦尔，\） ，其中一个 ∈ 乙（ħ）。特别是，如果A为次正规的，则通过用Kantorovich常数KK（⋅，⋅）细化Young不等式，我们证明\（w（A）\ le \ frac {1} {2 {\ operatorname {inf} } _ {\ left \ Vert x \ right \ Vert = 1} \ zeta（x）} \ left | \ left \ Vert A \ right。\ right | ++ \ left | \ left。{A} ^ {\ ast } \ right \ Vert \ right | \ le \ frac {1} {2} \ left | \ left \ Vert A \ right。\ right | + \ left | \ left。{A} ^ {\ ast} \ right \ Vert \ right |，\）其中\（\ upzeta（x）= K {\ left（\ frac {\ left \ langle \ left | A \ right | x，x \ right \ rangle} {\ left \ langle \ left | {A} ^ {\ ast } \ right | x，x \ right \ rangle}，2 \ right）} ^ r，r = \ min \ left \ {\ uplambda，1- \ uplambda \ right \}，\）和0≤λ≤1。我们还给出了经典的数值半径功率不等式反向瓦特（甲^ñ）≤ 瓦特^ñ（甲），用于操作者的任何一个 ∈ 乙（ħ）情况下ñ = 2 。