Refinements of Some Reverses of Schwarz’s Inequality in 2−inner Product Spaces and Applications for Integrals

The concepts of 2−inner products and 2−inner product spaces have been intensively studied by many authors in the last three decades. A systematic presentation of the recent results related to the theory of 2−inner product spaces as well as an extensive list of the related references can be found in the book [5]. We recall here the basic definitions and the elementary properties of 2−inner product spaces that will be used in the sequel (see also [3]). Let X be a linear space of dimension greater than 1 over the number field K, when K = R or K = C. Suppose that (·, ·|·) is a K-valued function defined on X ×X ×X satisfying the following conditions: (2I1) (x, x|z) ≥ 0 and (x, x|z) = 0 if and only if x and z are linearly dependent, (2I2) (x, x|z) = (z, z|x) , (2I3) (y, x|z) = (x, y|z), (2I4) (αx, y|z) = α (x, y|z) for any scalar α ∈ K, (2I5) (x+ x , y|z) = (x, y|z) + (x, y|z) , where x, x, y, z ∈ X. The functional (·, ·|·) is called a 2−inner product on X and (X, (·, ·|·)) is called a 2−inner product space (or 2-pre-Hilbert space) [5]. Some basic properties of the 2−inner product spaces can be immediately obtained as follows: (1) If K = R, then (2I3) reduces to (y, x|z) = (x, y|z) . (2) From (2I3) and (2I4) , we have (0, y|z) = (x, 0|z) = 0