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Quasi Gruss Type Inequalities for Complex Functions Dened on Unit Circle with Applications for Unitary Operators in Hilbert Spaces

The concept of Riemann-Stieltjes integral ∫ b a f(t) du(t), where f is called the integrand and u is called the integrator, plays an important role in Mathematics, for instance in the definition of complex integral, the representation of bounded linear functionals on the Banach space of all continuous functions on an interval [a, b], in the spectral representation of selfadjoint operators on complex Hilbert spaces and other classes of operators such as the unitary operators, etc. One can approximate the Riemann-Stieltjes integral ∫ b a f(t) du(t) with the following simpler quantity: