Difference between revisions of "Delta Minkowski inequality"
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==Theorem== | ==Theorem== | ||
Let $a,b \in \mathbb{T}$ and $p>1$. For [[continuity | rd-continuous]] $f,g \colon [a,b] \cap \mathbb{T} \rightarrow \mathbb{R}$ we have | Let $a,b \in \mathbb{T}$ and $p>1$. For [[continuity | rd-continuous]] $f,g \colon [a,b] \cap \mathbb{T} \rightarrow \mathbb{R}$ we have | ||
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{{:Delta inequalities footer}} | {{:Delta inequalities footer}} | ||
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Latest revision as of 00:38, 15 September 2016
Theorem
Let $a,b \in \mathbb{T}$ and $p>1$. For rd-continuous $f,g \colon [a,b] \cap \mathbb{T} \rightarrow \mathbb{R}$ we have $$\left( \displaystyle\int_a^b |(f+g)(t)|^p \Delta t \right)^{\frac{1}{p}} \leq \left( \displaystyle\int_a^b |f(t)|^p \Delta t \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(t)|^p \Delta t\right)^{\frac{1}{p}}.$$
Proof
References
Ravi Agarwal, Martin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 3.3
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
