I find the standard way of defining convex and concave functions slightly tricky. For me, it is the introduction of the new variable $0<λ<1$. However, I understand it intuitively. I was wondering if this definition is equivalent.
Let $f:[a,b] \rightarrow \mathbb{R}$ be a real function. We say $f$ is convex if for all $m,n \in [a,b]$ and for all $x \in (m,n)$ we have $$f(x) \leq \frac{x-m}{n-m}(f(n) - f(m)) + f(m) $$ If the inequality is reversed, we say $f$ is concave .
If you require $n>m$ (which may be implicit in your reference to $x\in (m,n)$), then it looks like you've got equivalence between the standard definition and your new one, $\lambda=(x-m)/(n-m)$. Caveat: I'm not sure your technique is any simpler.