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

$$m\le u(w)\qquad\text{and}\qquad u(w)\le M\qquad\text{for every }w\in D.$$

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

$$y=\lambda\,z+(1-\lambda)\,x .$$

Then

$$u(y)-u(x)\le\lambda\,(M-m).$$