Demonstrate that for a concave downward function that the value of the derivative $\frac{dy}{dx}$ is always higher than $\frac{y}{x}$

Question

I would like to demonstrate that for a concave downward function passing through the origin that the value of the derivative $\frac{dy}{dx}$ is always higher than $\frac{y}{x}$. As shown in figure, it is simple to understand graphically, but for a generical concave downward function, without having a specific function, I don't know how to start.

Thank you in advance

Answer 1

Let's use $$y=-(x-2)^2+C$$ Then at $x=1$ one gets $$\frac yx=C-1$$ and $$\frac{dy}{dx}=-2(1-2)=2$$ Notice that you can change the $y/x$ ratio to any number, while the derivative is always constant. So you can't prove that it is smaller or larger.

EDIT With the new addition to the question that the function passes through origin

The definition of concave function $f$ is that $\forall \lambda\in(0,1)$ $$f((1-\lambda)a+\lambda b)\ge(1-\lambda)f(a)+\lambda f(b)$$ So let's choose $a=0$ and $b=x$. We know that $f(0)=0$, so $$f(\lambda x)\ge\lambda f(x)$$ Then we want to calculate $$\frac{f(x)-f(x-h)}h$$ We can choose $x-h=\lambda x$ or $h=(1-\lambda)x$. $$\frac{f(x)-f(x-h)}h=\frac{f(x)-f(\lambda x)}{(1-\lambda)x}\le\frac{f(x)-\lambda f(x)}{(1-\lambda)x}=\frac{f(x)}{x}$$