# Delta Opial inequality

From timescalewiki

## 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 |