Reverses of the first Hermite-Hadamard type inequality for the square operator modulus in Hilbert spaces

‎Let (H;⟨⋅‎,‎⋅⟩) be a complex‎ ‎Hilbert space‎. ‎Denote by B(H) B(H) the Banach C^∗ -‎algebra of bounded linear operators on H ‎. ‎For A∈B(‎H) we define the modulus of A by |A|‎:‎=(‎A∗A)1/2 and \ funcReA:=12(A∗‎‎+A)‎.‎ .‎ In this paper we show among other that‎, ‎if A, B∈‎‎B(H) with 0≤m≤|(1−t)‎‎A+tB|^2≤M for all t∈[0,1]‎,‎ then ‎‎0‎‎≤∫1_0f(|(1−t)A+tB|‎‎2)dt−f(|A|2+\funcRe(‎‎B∗A)‎+‎|B|23)≤2[f(m)‎+‎f(M)2−f(‎m+M2)]1H‎‎ ‎‎ ‎for operator convex functions f:[0,∞)→R ‎. ‎Applications for power and logarithmic functions are also provided‎