Mathematics Subject Classification: 26D10.The sharp constant is obtained for the Hardy-Stein-Weiss inequality for fractional Riesz potential operator in the space L^p(R^n, ρ) with the power weight ρ = |x|^β. As a corollary, the sharp constant is found for a similar weighted inequality for fractional powers of the Beltrami-Laplace operator on the unit sphere.

Some new bounds for the Čebyšev functional in terms of the Lebesgue norms $${∥f-\frac{1}{b-a}{\int }_a^{b}f\left(t\right)\mathrm{d}t∥}_{[a,b],p}$$ and the $\Delta$-seminorms $${\parallel f\parallel }_p^{\Delta }:={\left({\int }_a^b{\int }_a^b{|f\left(t\right)-f\left(s\right)|}^p\mathrm{d}t\mathrm{d}s\right)}^{\frac{1}{p}}$$ are established. Applications for mid-point and trapezoid inequalities are provided as well.