Diamond alpha Jensen's inequality

Theorem: Let $\mathbb{T}$ be a time scale with $a,b \in \mathbb{T}$ with $a<b$ and $c,d \in \mathbb{R}$. If $g \colon [a,b]\cap\mathbb{T} \rightarrow (c,d)$ is continuous and $f \colon (c,d) \rightarrow \mathbb{R}$ is convex, then $$f \left( \dfrac{\int_a^bg(s) \Diamond_{\alpha}s}{b-a} \right) \leq \dfrac{\int_a^bf(g(s))\Diamond_{\alpha}s}{b-a}.$$

Proof: █

References

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