Several inequalities for an integral transform of positive operators in Hilbert spaces with applications

For a continuous and positive function \(w\left( \lambda \right) ,\) \(\lambda>0\) and \(\mu \) a positive measure on \((0,\infty )\) we consider the following Integral Transform \[ \begin{equation*} \mathcal{D}\left( w,\mu \right) \left( T\right) :=\int_{0}^{\infty }w\left(\lambda \right) \left( \lambda +T\right)^{-1}d\mu \left( \lambda \right) , \end{equation*} \] where the integral is assumed to exist for \(T\) a postive operator on a complex Hilbert space \(H\). We show among others that, if \( \beta \geq A \geq \alpha > 0, \, B > 0 \) with \( M \geq B-A \geq m > 0 \) for some constants \( \alpha,  \beta,  m,  M \), then \[ \begin{align*} 0 & \leq \frac{m^{2}}{M^{2}}\left[ \mathcal{D}\left( w,\mu \right) \left(\beta\right) - \mathcal{D}\left( w,\mu \right) \left(M+\beta\right) \right] \\ & \leq \frac{m^{2}}{M}\left[ \mathcal{D}\left( w,\mu \right) \left(\beta\right) - \mathcal{D}\left( w,\mu \right) \left(M+\beta\right) \right] \left( B-A\right)^{-1} \\ & \leq \mathcal{D}\left( w,\mu \right) \left(A\right) - \mathcal{D}\left(w,\mu\right) \left(B\right) \\ & \leq \frac{M^{2}}{m}\left[ \mathcal{D}\left( w,\mu \right) \left(\alpha\right) - \mathcal{D}\left( w,\mu \right) \left(m+\alpha\right) \right] \left(B-A\right)^{-1} \\ & \leq \frac{M^{2}}{m^{2}}\left[ \mathcal{D}\left( w,\mu \right) \left(\alpha\right) - \mathcal{D}\left( w,\mu \right) \left(m+\alpha\right) \right]. \end{align*} \] Some examples for operator monotone and operator convex functions as well as for integral transforms \(\mathcal{D}\left( \cdot ,\cdot \right) \) related to the exponential and logarithmic functions are also provided.