🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

Let $a,b\in \mathbb{R}$ such that $a\ge 1$ and $b>1$ and let $T,f:{\mathbb{N}}^0\to {\mathbb{R}}^{\ge 0}$ be defined such that $${\displaystyle T\left(n\right):=aT\left(\frac{n}{b}\right)+f\left(n\right)}$$ Then the following statements are true

• If $f\in \mathcal{O}\left(n^{{\log }_{b}a-ϵ}\right)$ for some $ϵ\in {\mathbb{R}}^{>0}$, then $T\left(n\right)\in \Theta \left(n^{{\log }_{b}a}\right)$
• If $f\in \Theta \left(n^{\log ba}\right)$ then $T\left(n\right)\in \Theta \left(n^{\log ba}\ln \left(n\right)\right)$
• If $f\in \Theta \left(n^{{\log }_b\left(a\right)+ϵ}\right)$ for some constant $ϵ\in {\mathbb{R}}^{>0}$ and the function $f^{\prime }\left(n\right):=af\left(\frac{n}{b}\right)$ is eventually dominated up to a constant $c\in \mathbb{R}$ by $cf$ where $c<1$, then $T\left(n\right)\in \Theta \left(f\right)$