Improvements of Berezin number inequalities

In this paper, we generalize several Berezin number inequalities involving product of operators, which acting on a Hilbert space H (Omega). Among other inequalities, it is shown that if A,B are positive operators and X is any operator, then ber(r)(H-alpha(A, B)) <= parallel to X parallel to(r)/2 (ber(A(r) + B-r) - 2 inf(parallel to(k) over cap lambda parallel to=1) eta((k) over cap (lambda)) <= parallel to X parallel to(r)/2 (ber(aA(r) + (1 - alpha)B-r) +ber((1 - alpha)A(r) +alpha B-r) - 2 inf(parallel to(k) over cap lambda parallel to=1) eta((k) over cap (lambda))), where eta((k) over cap (lambda)) = r(0)(A(r)(k) over cap (lambda)(k) over cap (lambda))(1/2) - (B-r(k) over cap (lambda)(k) over cap (lambda))(1/2))(2), r >= 2, 0 <= alpha <= 1, r(0) = min{alpha, 1 - alpha} and H-alpha(A, B) = (A(alpha)XB(1-alpha) + A(1-alpha)XB(alpha))/2.