I’m self-studying real analysis using A course in mathematical analysis by Garling

One of the problems after the subsection on convex functions defines a function as midpoint convex if $$f(frac{a+b}{2}) leq frac{f(a)+f(b)}{2}$$

The problem is then to prove that, if f is midpoint convex, $hgt 0$ and $n in Bbb{N}$$$frac{f(c-h)-f(c)}{n+1} leq f(c+frac{h}{n}) – f(c)$$

I don’t think this statement is true. Let $f(x) = x^2,$ $n = 1,$ $ c = -1$ and $h = 0.25$

f(x) is midpoint convex as $$frac{a^2+2ab+b^2}{4} leq frac{a^2+b^2}{2}$$ is equivalent to

$$2ab leq a^2+b^2$$ which is true but $$frac{f(-1.25)-f(-1)}{2} = 0.28125 gt -0.4375 = f(-0.75) – f(1)$$