Refinements of norm and numerical radius inequalities

Several refinements of norm and numerical radius inequalities bounded linear operators on a complex Hilbert space are given. In particular, we show that if A is operator space, then 14A*A+AA*≤18A+A*2+A−A*2+c2(A+A*)+c2(A−A*)≤w2(A) 12A∗A+AA∗−14(A+A∗)2(A−A∗)212≤w2(A)≤12A∗A+AA∗, where ∥⋅∥, w(⋅) c(⋅) the norm, Crawford number, respectively. Further, prove A, D AD∗≤∫01(1−t)(A2+D2)/2+tAD∗I2dt12≤12A2+D2, |A|2=A∗A |D|2=D∗D. This refinement well-known inequality obtained by Bhatia Kittaneh.