Geodesic
A refinement of the Cauchy-Schwarz inequality accompanied by new numerical radius upper bounds
Filomat, Tome 37 (2023) no. 3, p. 971

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DOI

This present work aims to ameliorate the celebrated Cauchy-Schwarz inequality and provide several new consequences associated with the numerical radius upper bounds of Hilbert space operators. More precisely, for arbitrary a, b ∈ H and α ≥ 0, we show that |〈a, b〉|2 ≤ 1 α + 1 ∥a∥ ∥b∥ |〈a, b〉| + α α + 1 ∥a∥2∥b∥2 ≤ ∥a∥2∥b∥2. As a consequence, we provide several new upper bounds for the numerical radius that refine and generalize some of Kittaneh’s results in [A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math. 2003;158:11-17] and [Cauchy-Schwarz type inequalities and applications to numerical radius inequalities. Math. Inequal. Appl. 2020;23:1117-1125], respectively. In particular, for arbitrary A, B ∈ B(H) and α ≥ 0, we show the following sharp upper bound w2 (B∗A) ≤ 1 2α + 2 | A|2 + | B|2 w(B∗A) + α 2α + 2 | A|4 + | B|4 , with equality holds when A = B = 0 1 0 0. It is also worth mentioning here that some specific values of α ≥ 0 provide more accurate estimates for the numerical radius. Finally, some related upper bounds are also provided.
DOI : 10.2298/FIL2303971A
Classification : 47A12, 47A63 47A30, 47B33
Keywords: Numerical radius, Usual operator norm 
Mohammed Al-Dolat; Imad Jaradat. A refinement of the Cauchy-Schwarz inequality accompanied by new numerical radius upper bounds. Filomat, Tome 37 (2023) no. 3, p. 971 . doi: 10.2298/FIL2303971A
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     title = {A refinement of the {Cauchy-Schwarz} inequality accompanied by new numerical radius upper bounds},
     journal = {Filomat},
     pages = {971 },
     year = {2023},
     volume = {37},
     number = {3},
     doi = {10.2298/FIL2303971A},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2303971A/}
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