In this paper we point out a trapezoid inequality for the Riemann-Stieltjes integral , where is of bounded variation on [] and is of —-Hölder type on [] . Applications to the approximation problem of Riemann-Stieltjes integrals in terms of Riemann-Stieltjes sums are also given

Some new reverses of Bessel's inequality for orthonormal families in real or complex inner product spaces are pointed out. Applications for some Grüss type inequalities and for determinantal integral inequalities are given as well

A new counterpart of Bessel's inequality for orthornormal families in real or complex 2-inner product spaces is obtained. Applications for some Grüss type results with applications for determinantal integral inequalities are also provided

Refinements of some recent reverse inequalities for the celebrated Cauchy-Bunyakovsky-Schwarz inequality in 2—inner product spaces are given. Using this framework, applications for determinantal integral inequalities are also provided

Norm estimates are developed between the Bochner integral of a vector-valued function in Banach spaces having the Radon-Nikodym property and the convex combination of function values taken on a division of the interval []

Some results related to the Bombieri type generalisation of Bessel's inequality in 2-inner product spaces are given. The corresponding versions for Selberg and Heilbronn inequalities for 2-inner products and applications for determinantal integral inequalities are also pointed out