Document Type : Research Paper

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Abstract

We obtain some inequalities related to the powers of numerical radius inequalities of Hilbert space operators. Some results that employ the Hermite-Hadamard inequality for vectors in normed linear spaces are also obtained. We improve and generalize some inequalities with respect to Specht's ratio. Among them, we show that, if $A, B\in \mathcal{B(\mathcal{H})}$ satisfy in some conditions, it follows that
\begin{equation*}
\omega^2(A^*B)\leq
\frac{1}{2S(\sqrt{h})}\Big\||A|^{4}+|B|^{4}\Big\|-\displaystyle{\inf_{\|x\|=1}}
\frac{1}{4S(\sqrt{h})}\big(\big\langle \big(A^*A-B^*B\big)
x,x\big\rangle\big)^2
\end{equation*}
for some $h>0$, where $\|\cdot\|,\,\,\,\omega(\cdot)$ and $S(\cdot)$ denote the usual operator norm, numerical radius and the Specht's ratio, respectively.

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