Ostrowski’s inequality on time scales

A time scale version of Ostrowski’s inequality is given as follows: Let f,g∈Crd([a,b],R) be two linearly independent functions, then for any α∈[−1,1]α∈[−1,1] and any arbitrary x∈Crd([a,b],R)x∈Crd([a,b],R) with equation(C1)∫abf(t)x(t)Δt=0,∫abg(t)x(t)Δt=1, the function αx(t)+(1−α)y(t),t∈[a,b] satisfies condition (C1) and ∫abx2(t)Δt≥∫ab[αx(t)+(1−α)y(t)]2Δt≥AAB−C2, where A=∫abf2(t)Δt,B=∫abg2(t)Δt,C=∫abf(t)g(t)Δt and y(t)=Ag(t)−Bf(t)AB−C2on [a,b].