Difference between revisions of "Delta Opial inequality"
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| − | + | __NOTOC__ | |
| − | + | ==Theorem== | |
| + | For a differentiable $x \colon [0,h] \cap \mathbb{T} \rightarrow \mathbb{R}$ with $x(0)=0$ we have | ||
$$\displaystyle\int_0^h |(x+x^{\sigma})x^{\Delta}|(t) \Delta t \leq h \displaystyle\int_0^h |x^{\Delta}|^2(t) \Delta t,$$ | $$\displaystyle\int_0^h |(x+x^{\sigma})x^{\Delta}|(t) \Delta t \leq h \displaystyle\int_0^h |x^{\Delta}|^2(t) \Delta t,$$ | ||
with equality when $x(t)=ct$. | with equality when $x(t)=ct$. | ||
| − | + | ||
| − | + | ==Proof== | |
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| − | |||
==References== | ==References== | ||
| − | + | {{PaperReference|Inequalities on Time Scales: A Survey|2001|Ravi Agarwal|author2 = Martin Bohner| author3 = Allan Peterson|prev=findme|next=findme}}: Theorem 6.1 | |
{{:Delta inequalities footer}} | {{:Delta inequalities footer}} | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 00:38, 15 September 2016
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
Ravi Agarwal, Martin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 6.1
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
