I'm an undergraduate student that find a problem for proofing theorem below. First it defined that: A function f defined on an interval (a,b), $-∞ ≤ a < b≤ ∞$, is said to be a convex function if for all x,y in (a,b) and for all $0 < γ < 1$,
$$f[γx + (1 - γ)y] ≤ γf(x) + (1 - γ)f(y)$$ We say f is strictly convex if the above inequality is strict.
The theorem said that, if f is differentiable on (a,b), then
1) f is convex if and only if $f' (x) ≤ f'(y)$, for all $a < x < y < b$,
2) f is strictly convex if and only if $f'(x) < f'(y)$, for all $a < x < y < b$.
I try to proof the first part from the left to the right, in this way:
a) Let f is differentiable on (a,b), and f is convex. Let $γ=\frac{(k-j)}{(k-x)}$. The value of γ is arbitrary value such that $0<γ<1$. Choosing x,j,k and y, such that $a < x <j<k< y < b$, and $j=\frac{(k-j)}{(k-x)} x+\frac{(j-x)}{(k-x)}k$, so, by the definiton of convex $$f(j)=f\left(\frac{(k-j)}{(k-x)} x+\frac{(j-x)}{(k-x)} k\right)≤\frac{(k-j)}{(k-x)} f(x)+\frac{(j-x)}{(k-x)} f(k)$$ Then
$(k-x)f(j)≤(k-j)f(x)+(j-x)f(k)$
$((k-j)+(j-x) )f(j)≤(k-j)f(x)+(j-x)f(k)$ $(k-j)f(j)+(j-x)f(j)≤(k-j)f(x)+(j-x)f(k)$ $(k-j)f(j)-(k-j)f(x)≤(j-x)f(k)-(j-x)f(j)$ $(k-j)[f(j)-f(x) ]≤(j-x)[f(k)-f(j) ]$
$$\frac{[f(j)-f(x) ]}{(j-x)}≤\frac{[f(k)-f(j) ]}{(k-j)}$$ Doing the same step for j,k and y, yields $$\frac{[f(k)-f(j) ]}{(k-j)}≤\frac{[f(y)-f(k) ]}{(y-k)}$$ So, we get $$\frac{[f(j)-f(x) ]}{(j-x)}≤\frac{[f(k)-f(j) ]}{(k-j)}≤\frac{[f(y)-f(k) ]}{(y-k)}$$ $$\frac{[f(j)-f(x) ]}{(j-x)}≤\frac{[f(y)-f(k) ]}{(y-k)}$$
and the last step i was taking the limit j→x for the left side and k→y for the right one, yields
$ f' (x)=lim_{j→x}\left(\frac{[f(j)-f(x) ]}{(j-x)}\right)≤lim_{k→y}\left(\frac{[f(y)-f(k) ]}{(y-k)}\right)=f' (y) $
is it reasonable to doing this? i have searching for the proof in some reference but i haven't find a clear explanation yet. ones said that the last step you should take the limit $j→x^+$ and $k→y^-$, but i didn't understand what the reason. I find this problem in Robert V. Hogg math statistic book. The right reference should be analysis book from Hewitt-Stromberg but i didn't find it in my library.