Difference between revisions of "Diamond alpha Hölder inequality"
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| + | <strong>Theorem:</strong> Let $\mathbb{T}$ be a [[time scale]] with $a,b \in \mathbb{T}$, $a<b$,and $f,g \colon [a,b]\cap\mathbb{T} \rightarrow [0,\infty)$. Also assume that | ||
| + | $$\displaystyle\int_a^b h(x)g^q(x)\Diamond_{\alpha} x >0,$$ | ||
| + | where $q$ obeys $\dfrac{1}{p}+\dfrac{1}{q}=1$ with $p>1$. Then, | ||
| + | $$\displaystyle\int_a^b h(x)f(x)g(x) \Diamond_{\alpha} x \leq \left( \displaystyle\int_a^b h(x) f^p(x) \Diamond_{\alpha} x \right)^{\frac{1}{p}} \left( \displaystyle\int_a^b h(x)g^q(x) \Diamond_{\alpha} x \right)^{\frac{1}{q}}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | =References= | ||
| + | [http://arxiv.org/pdf/0712.1680.pdf] | ||
Latest revision as of 12:36, 28 March 2015
Theorem: Let $\mathbb{T}$ be a time scale with $a,b \in \mathbb{T}$, $a<b$,and $f,g \colon [a,b]\cap\mathbb{T} \rightarrow [0,\infty)$. Also assume that $$\displaystyle\int_a^b h(x)g^q(x)\Diamond_{\alpha} x >0,$$ where $q$ obeys $\dfrac{1}{p}+\dfrac{1}{q}=1$ with $p>1$. Then, $$\displaystyle\int_a^b h(x)f(x)g(x) \Diamond_{\alpha} x \leq \left( \displaystyle\int_a^b h(x) f^p(x) \Diamond_{\alpha} x \right)^{\frac{1}{p}} \left( \displaystyle\int_a^b h(x)g^q(x) \Diamond_{\alpha} x \right)^{\frac{1}{q}}.$$
Proof: █
