For a continuous and positive function w(λ), λ 0 and μ a positive measure on (0, ∞) we consider the following monotonic integral transform where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among others that, if β ≥ A, B ≥ α 0, and 0 δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, then and where  is the second derivative of as a real function. Applications for power function and logarithm are also provided.