Ostrowski-Type Inequalities in Complex Hilbert Spaces for Square Modulus Convex Functions of Operators

The present work is devoted to deriving Ostrowski-type inequalities for square modulus convex mappings K:[α,β]⊂R→B(H) in complex Hilbert spaces. We show that the deviation of the weighted integral mean ∫Xρ|K∘φ|2dμ from the pointwise value |K(ξ)|2 at any point ξ∈[α,β] is bounded in terms of the weighted standard deviation σρ(φ) of φ, giving the operator-valued analogue of the classical Ostrowski inequality. Taking ξ=α+β2 yields a midpoint inequality with an explicit constant. We also obtain a distribution-dependent estimate for the Jensen gap that vanishes whenever φ is constant. New two-point and multi-point Ostrowski inequalities and a superadditivity property of the Jensen gap are also derived.