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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 10, pp. 1443–1451, October, 2020. Ukrainian DOI: 10.37863/umzh.v72i10.6027.
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Heydarbeygi, Z., Amyari, M. & Khanehgir, M. Some Refinements of Numerical Radius Inequalities. Ukr Math J 72, 1664–1674 (2021). https://doi.org/10.1007/s11253-021-01879-1
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DOI: https://doi.org/10.1007/s11253-021-01879-1