Increment Bound for a Convex Function through an Extended Point

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, let $C$ be a convex subset of Euclidean space $\mathbb{R}^{n}$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and let $u:C\to\mathbb{R}$ be convex on $C$.

Let $D\subseteq C$ and let $m,M\in\mathbb{R}$ satisfy

Let $x,y,z\in D$ and let $\lambda\in\mathbb{R}$ satisfy $0\le\lambda$, $\lambda\le1$ and

Then