Upper and lower bounds for the numerical radius with an application to involution operators

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More accurate numerical radius inequalities (I) (2019, background): “|x,xi) = f(h(t|A|+(1−t)|A∗|)x,xi) ≤ hf(t|A|+(1− t)|A∗|)x,xi (by Lemma 1.2) ≤ thf(|A|)x,xi+(1− t)hf(|A∗|)x,xi, where the last inequality follows from operator convexity of f. Now, integrating over ton [0,1], (2.3) Z 1 0 f(th|A|x,xi +(1− t)h|A∗|x,xi) ≤ ˝Z 1 f(t|A|+(1− t)|A∗|)dtx,x ˛ ≤ 1 2 hf((|A|)+f(|”
Numerical radius inequalities and its applications in estimation of zeros of polynomials (2019, background): “Abu-Omar and Kittaneh [1] also proved that this bound is better than the bounds obtained in [13, 12] w(T ) ≤ 1 2 ( ‖T ‖+ ‖T (2)‖ 1 2 )”
A-numerical radius inequalities for semi-Hilbertian space operators (2019, unknown): “In Section 2, inspired by the numerical radius inequalities of bounded linear operators in [1], [2], [21], [23], [33] and by using some ideas of them, we first state a useful characterization of the A-numerical radius for T ∈ BA(H) as follows: wA(T ) = sup α(2)+β(2)=1 ∥ ∥ ∥ ∥ α T + T ♯A 2 + β T − T ”