Good day to everyone. I am interested in the geometric intuition for the following statement:
Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and $c,d \in \mathbb{R}$. Then for any $y \in \mathbb{R}$ we have \begin{equation} \Big(f(y) - c\Big) \Big( y - d \Big) \geq \Big(f(d) - c\Big) \Big( f^{-1}(c) - d \Big). \end{equation}
It is a lemma which I have encountered in the article (Lemma 3.1):
T. Kerkhoven and J. W. Jerome. $L_\infty$ stability of finite element approximations of elliptic gradient equations. Numerische Mathematik, 57:561, 1990.
The lemma is proven in the paper. The proof is analytic and it is rather elementary. However, it seems to me that this statement is releated to some geometric property of graphs of real increasing functions. And I am interested in this geometric intuition. Perhaps it is a standard argument, but I cannot see it. Thus my question is:
What are the geometrical reasons for the statement cited above?
Some thoughts:
It is maybe relevant to multiply the inequality by (-1), reversing its direction. It is so, because if we take $f(x):=x$, then also $f^{-1}(x) = x$ and we obtain $LHS:=(y-c)(y-d) \geq -(d-c)^2=:RHS$, so RHS is negative for all $c \neq d$. Thus if LHS is positive, then the signs differ and probably no geometrical argument can be given. So the interesting case would be when LHS is also negative.