Diamond alpha Minkowski's inequality

Theorem: Let $\mathbb{T}$ be a time scale with $a,b \in \mathbb{T}$, $a<b$ and $p>1$. For continuous functions $f,g \colon [a,b]\cap \mathbb{T}\rightarrow \mathbb{R}$ we have $$\left( \displaystyle\int_a^b |(f+g)(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}\leq \left( \displaystyle\int_a^b |f(x)|^p\Diamond_{\alpha}x \right)^{\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}}.$$

Proof: █