Difference between revisions of "Diamond alpha Minkowski's inequality"
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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 | |
$$\left( \displaystyle\int_a^b |(f+g)(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}\leq \left( \displaystyle\int_a^b |f(x)|^p\Diamond_{\alpha}x \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}.$$ | $$\left( \displaystyle\int_a^b |(f+g)(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}\leq \left( \displaystyle\int_a^b |f(x)|^p\Diamond_{\alpha}x \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}.$$ | ||
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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 $$\left( \displaystyle\int_a^b |(f+g)(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}\leq \left( \displaystyle\int_a^b |f(x)|^p\Diamond_{\alpha}x \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}.$$
Proof: █
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