Difference between revisions of "Delta Opial inequality"

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__NOTOC__
<strong>Theorem:</strong> For a differentiable $x \colon [0,h] \cap \mathbb{T} \rightarrow \mathbb{R}$ with $x(0)=0$ we have
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==Theorem==
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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 $x(t)=ct$.  
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<strong>Proof:</strong> █
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==Proof==
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==References==
 
==References==
[http://www.math.unl.edu/~apeterson1/pub/ineq.pdf R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey]
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{{PaperReference|Inequalities on Time Scales: A Survey|2001|Ravi Agarwal|author2 = Martin Bohner| author3 = Allan Peterson|prev=findme|next=findme}}: Theorem 6.1
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{{:Delta inequalities footer}}
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[[Category:Theorem]]
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[[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 AgarwalMartin 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