Diamond alpha Hölder inequality

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: █

References

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