Some improvements of numerical radius inequalities via Specht’s ratio

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 $A, Bin mathcal{B(mathcal{H})}$ 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(biglangle big(A^*A-B^*Bbig) x,xbigranglebig)^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.