New inequalities bound the square of the numerical radius w(A) by the norm of an integral average of (t|A| + (1-t)|A*|) squared, itself bounded by half the norm of |A| squared plus |A*| squared.
Kittaneh, Notes on some inequalities for Hilbert space operators , Publ
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More accurate numerical radius inequalities
New inequalities bound the square of the numerical radius w(A) by the norm of an integral average of (t|A| + (1-t)|A*|) squared, itself bounded by half the norm of |A| squared plus |A*| squared.