In Aleksandrov and Peller (2010) [2] we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in Aleksandrov and Peller (2010) [2] for certain special classes of functions. In particular, we improve estimates of Kato (1973) [18] and show that. |||S|-|T|||≤C||S-T||log(2+log||S||+||T||/||S-T||) for all bounded operators S and T on Hilbert space. Here |S|=def(S*S)^1/2. Moreover, we show that this inequality is sharp. We prove in this paper that if f is a nondecreasing continuous function on R that vanishes on (-∞,0] and is concave on [0,∞), then its operator modulus of continuity Ωf admits the estimate. Ω_f(delta;)≤const ∫_e^∞ f(δt)dt/t²logt, δ>0.We also study the problem of sharpness of estimates obtained in Aleksandrov and Peller (2010) [2,3]. We construct a C∞ function f on R such that ||f||L||∞1, ||f||Lip||≤1, and. In the last section of the paper we obtain sharp estimates of ||f(A)-f(B)|| in the case when the spectrum of A has n points. Moreover, we obtain a more general result in terms of the ε-ntropy of the spectrum that also improves the estimate of the operator moduli of continuity of Lipschitz functions on finite intervals, which was obtained in Aleksandrov and Peller (2010) [2].