Delta Opial inequality

Theorem: For a differentiable $x \colon [0,h] \cap \mathbb{T} \rightarrow \mathbb{R}$ with $x(0)=0$ we have $$\displaystyle\int_0^h |(x+x^{\sigma})x^{\Delta}|(t) \Delta t \leq h \displaystyle\int_0^h |x^{\Delta}|^2(t) \Delta t,$$ with equality when $x(t)=ct$.

Proof: █

References

R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey

$\Delta$-Inequalities

Bernoulli

Bihari

Cauchy-Schwarz

Gronwall

Hölder

Jensen

Lyapunov

Markov

Minkowski

Opial

Tschebycheff

Wirtinger