Difference between revisions of "Diamond alpha Minkowski's inequality"
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(Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $\mathbb{T}$ be a time scale with $a,b \in \mathbb{T}$, $a<b$ and $p...") |
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| − | + | ==Theorem== | |
| − | + | If $\mathbb{T}$ is a [[time scale]], $a,b \in \mathbb{T}$ with $a<b$, $p>1$, and $f,g \colon [a,b]\cap \mathbb{T}\rightarrow \mathbb{R}$ are continuous, then | |
| − | $$\left( \displaystyle\int_a^b |(f+g)(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}\leq \left( \displaystyle\int_a^b |f(x)|^p\Diamond_{\alpha}x \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}} | + | $$\left( \displaystyle\int_a^b |(f+g)(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}\leq \left( \displaystyle\int_a^b |f(x)|^p\Diamond_{\alpha}x \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}},$$ |
| − | + | where $\displaystyle\int \ldots \Diamond_{\alpha} x$ denotes the [[diamond alpha integral]]. | |
| − | + | ||
| − | + | =References= | |
| − | + | [http://arxiv.org/pdf/0712.1680.pdf] | |
Latest revision as of 15:17, 21 January 2023
Theorem
If $\mathbb{T}$ is a time scale, $a,b \in \mathbb{T}$ with $a<b$, $p>1$, and $f,g \colon [a,b]\cap \mathbb{T}\rightarrow \mathbb{R}$ are continuous, then $$\left( \displaystyle\int_a^b |(f+g)(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}\leq \left( \displaystyle\int_a^b |f(x)|^p\Diamond_{\alpha}x \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}},$$ where $\displaystyle\int \ldots \Diamond_{\alpha} x$ denotes the diamond alpha integral.
