Difference between revisions of "Delta Hölder inequality"

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__NOTOC__
<strong>Theorem:</strong> Let $a,b \in \mathbb{T}$. For [[continuity | rd-continuous]] $f,g \colon [a,b]\cap\mathbb{T} \rightarrow \mathbb{R}$ we have
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==Theorem==
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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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<strong>Proof:</strong> █
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==Proof==
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==References==
 
==References==
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* {{PaperReference|Inequalities on Time Scales: A Survey|2001|Ravi Agarwal|author2 = Martin Bohner| author3 = Allan Peterson|prev=Minkowski's inequality for integrals|next=Sum rule for derivatives}}: $3.3.1$
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[http://www.math.unl.edu/~apeterson1/pub/ineq.pdf R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey]
 
[http://www.math.unl.edu/~apeterson1/pub/ineq.pdf R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey]
  
 
{{:Delta inequalities footer}}
 
{{:Delta inequalities footer}}

Revision as of 00:07, 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

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