Difference between revisions of "Delta Cauchy-Schwarz inequality"
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| − | <strong>Theorem:</strong> Let $a,b \in \mathbb{T}$. For rd-continuous $f,g \colon [a,b]\cap \mathbb{T} \rightarrow \mathbb{R}$ we have | + | <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 |
$$\displaystyle\int_a^b |f(t)g(t)| \Delta t \leq \sqrt{\left( \displaystyle\int_a^b |f(t)|^2 \Delta t \right) \left( \displaystyle\int_a^b |g(t)|^2 \Delta t \right)}$$ | $$\displaystyle\int_a^b |f(t)g(t)| \Delta t \leq \sqrt{\left( \displaystyle\int_a^b |f(t)|^2 \Delta t \right) \left( \displaystyle\int_a^b |g(t)|^2 \Delta t \right)}$$ | ||
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Revision as of 04:36, 6 September 2014
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
R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey
