SHARP OPERATOR MEAN INEQUALITIES OF THE NUMERICAL RADII

We present several sharp upper bounds and some extension for product operators. Among other inequalities, it is shown that if 0 < mI <= B* f(2) (vertical bar X vertical bar)B, A*g(2)(X-vertical bar*vertical bar) A <= MI, f, g are non-negative continuous functions on [0,8) such that f(t)g(t) = t, (t >= 0), then for all non-negative operator monotone decreasing function h on [0,infinity), we obtain that parallel to h(B* f(2)(vertical bar X vertical bar)B sigma h (A*g(2)(vertical bar X*vertical bar)parallel to <= mk/M h(vertical bar <(A*XB)x, x >vertical bar). As an application of the above inequality, it is shown that omega(A*XB) <= mk/M parallel to B*f(2)(vertical bar X vertical bar)BIA*g(2)(vertical bar X vertical bar)A parallel to where, k = (M+m)(2)/4mM and sigma is an operator mean s. t., ! <= sigma <= del.