Weighted Higher-Order Beesack–Opial Inequalities on Time Scales

This work constructs a family of weighted higher-order Opial-type estimates on an arbitrary time scale T, with the weight structure generated by a non-decreasing auxiliary function ω and its delta derivative ωΔ. For n-times delta differentiable functions whose delta derivatives of orders 0,1,…,n−1 vanish at the left endpoint, the mixed functional ∫abv(t)|u(t)|p|uΔn(t)|qΔt is bounded by expressions involving only the highest-order delta derivative uΔn. Three forms are obtained: a Hölder-type product estimate, a Young-type two-term estimate with a free balancing parameter θ>0, and an optimized single-integral estimate with the explicit constant K=(r−1)1−1/r/r, where r=(p+q)/q. The proofs rely on the Taylor representation on time scales, Hölder’s inequality, Jensen’s inequality, and Fubini’s theorem. The resulting bounds are governed by computable kernels that encode the interaction between the Taylor monomials, the auxiliary weight, and the external weight. Specializations to T=R, T=Z, and the quantum lattice q0N0 recover classical, discrete, weighted, and quantum Opial inequalities with their standard constants and also yield several higher-order weighted estimates. As an application, a uniqueness criterion for higher-order dynamic initial value problems is established.