I was reading a proof in which the author claimed that, for a function $f(x)$ on $(0,1)$ which is continuous and strictly increasing, the following condition implies that convexity holds.
Let $A > B \geq C > D$ and $A - B = C- D$. Then $f(A) - f(B) \geq f(C) - f(D)$.
Pictorially it makes intuitive sense to me that the above condition would imply convexity, but is there a formal proof of this? Conversely, if $f(A) - f(B) \leq f(C) - f(D)$, would that imply concavity of $f$? My intuition is yes, though would like to double check.
Edit: Answered my own question, let B = C = (A + D)/2 to obtain midpoint convexity.