We show amongst other that if $f,g:[a,b]{ ightarrow}mathbb{C}$ are two functions of bounded variation and such that the Riemann-Stieltjes integral $int_a^bf(t)dg(t)$ exists, then for any continuous functions $h:[a,b]{ ightarrow}mathbb{C}$, the Riemann-Stieltjes integral $int_{a}^{b}h(t)d(f(t)g(t))$ exists and $${int}_a^bh(t)d(f(t)g(t))={int}_a^bh(t)f(t)d(g(t))+{int}_a^bh(t)g(t)d(f(t))$$. Using this identity we then provide sharp upper bounds for the quantity $$|int_a^bh(t)d(f(t)g(t))|$$ and apply them for trapezoid and Ostrowski type inequalities. Some applications for continuous functions of selfadjoint operators on complex Hilbert spaces are given as well.