logarithmic functions

‎Let $left( H;leftlangle cdot‎ ,‎cdot rightrangle right)$ be a complex‎ ‎Hilbert space‎. ‎Denote by $mathcal{B}left( Hright)$ the Banach $C^{ast }$-‎algebra of bounded linear operators on $H$‎. ‎For $Ain mathcal{B}left(‎Hright)$ we define the modulus of $A$ by $leftvert Arightvert‎ :‎=left(‎A^{ast }Aright) ^{1/2}$ and $func{Re}A:=frac{1}{2}left( A^{ast‎‎}+Aright)‎.‎$ In this paper we show among other that‎, ‎if $A,$ $Bin mathcal{‎‎B}left( Hright)$ with $0leq mleq leftvert left( 1-tright)‎‎A+tBrightvert ^{2}leq M$ for all $tin left[ 0,1right]‎,‎$ then begin{align*}‎ ‎0& leq int_{0}^{1}fleft( leftvert left( 1-tright) A+tBrightvert‎‎^{2}right) dt-fleft( frac{leftvert Arightvert ^{2}+func{Re}left(‎‎B^{ast }Aright)‎ +‎leftvert Brightvert ^{2}}{3}right) \‎ ‎& leq 2left[ frac{fleft( mright)‎ +‎fleft( Mright) }{2}-fleft( frac{‎m+M}{2}right) right] 1_{H}‎ ‎end{align*} ‎for operator convex functions $f:[0,infty )rightarrow mathbb{R}$‎. ‎Applications for power and logarithmic functions are also provided‎.