Difference between revisions of "Delta Hölder inequality"
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| − | + | __NOTOC__ | |
| − | + | ==Theorem== | |
| + | Let $a,b \in \mathbb{T}$. For [[continuity | rd-continuous]] $f,g \colon [a,b]\cap\mathbb{T} \rightarrow \mathbb{R}$ we have | ||
$$\displaystyle\int_a^b |f(t)g(t)| \Delta t \leq \left( \displaystyle\int_a^b |f(t)|^p \Delta t \right)^{\frac{1}{p}} \left(\displaystyle\int_a^b |g(t)|^q \Delta t \right)^{\frac{1}{q}}$$ | $$\displaystyle\int_a^b |f(t)g(t)| \Delta t \leq \left( \displaystyle\int_a^b |f(t)|^p \Delta t \right)^{\frac{1}{p}} \left(\displaystyle\int_a^b |g(t)|^q \Delta t \right)^{\frac{1}{q}}$$ | ||
where $p>1$ and $q = \dfrac{p}{p-1}$. | where $p>1$ and $q = \dfrac{p}{p-1}$. | ||
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| − | + | ==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 3.1 | |
{{:Delta inequalities footer}} | {{:Delta inequalities footer}} | ||
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 00:36, 15 September 2016
Theorem
Let $a,b \in \mathbb{T}$. For rd-continuous $f,g \colon [a,b]\cap\mathbb{T} \rightarrow \mathbb{R}$ we have $$\displaystyle\int_a^b |f(t)g(t)| \Delta t \leq \left( \displaystyle\int_a^b |f(t)|^p \Delta t \right)^{\frac{1}{p}} \left(\displaystyle\int_a^b |g(t)|^q \Delta t \right)^{\frac{1}{q}}$$ where $p>1$ and $q = \dfrac{p}{p-1}$.
Proof
References
Ravi Agarwal, Martin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 3.1
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
