Difference between revisions of "Delta Cauchy-Schwarz inequality"
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==Theorem== | ==Theorem== | ||
Let $a,b \in \mathbb{T}$. For [[continuity | rd-continuous]] $f,g \colon [a,b]\cap \mathbb{T} \rightarrow \mathbb{R}$ we have | 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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{{:Delta inequalities footer}} | {{:Delta inequalities footer}} | ||
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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 $$\displaystyle\int_a^b |f(t)g(t)| \Delta t \leq \sqrt{\left( \displaystyle\int_a^b |f(t)|^2 \Delta t \right) \left( \displaystyle\int_a^b |g(t)|^2 \Delta t \right)}$$
Proof
References
Ravi Agarwal, Martin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 3.2
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
