Tensorial Simpson Type Inequalities of Self Adjoint Operators in Hilbert Space

Several Simpson 1/8 tensorial-type inequalities for self adjoint operators have been obtained, with variations depending on the conditions imposed on the function f:

||1/8[f(A)⊗1 + 6f (A⊗1 + 1⊗B/2) + 1⊗f(B)] − ∫01f (λ1⊗B + (1−λ)A⊗1)dλ|| ≤ 5||1⊗B − A⊗1||/32 ||f’||I, +∞.

The concept we now call a tensor was not originally named that way. When Josiah Willard Gibbs first described the idea in the late 19th century, he used the term dyadic. Today, mathematicians define a tensor as the mathematical embodiment of Gibbs’ initial concept.

Tensors and inequalities are natural partners due to the widespread use of inequalities across mathematics. These mathematical statements of comparison have a profound impact on numerous scientific disciplines. While many types of inequalities exist, some of the most significant include Jensen’s, Ostrowski’s, Hermite’Hadamard’s, and Minkowski’s inequalities.