I have a doubt about a step used in the proof of the following theorem (Rao and Ren, Theory of Orlicz Spaces):
Let $\phi:(a,b)\to\mathbb{R}$ a function. $\phi$ is convex if and only if for each closed subinterval $[c,d]\subset(a,b)$ we have
\begin{align*}
\phi(x)=\phi(c)+\int_{c}^{x}\varphi(t)dt \quad c<x<d
\end{align*}
where $\varphi$ is a monotone, nondecreasing and left continuous function. Also, $\phi$ has a left and a right derivative at each point of (a,b) and they are equal except perhaps for at most a countable number of points.
I ask if it's right the following inequality the book establishes?
\begin{align*}
(D^{+}\phi)(x) = \lim_{h\to 0^{+}}\frac{\phi(x+h)-\phi(x)}{h}\leq\frac{\phi(d)-\phi(c)}{d-c}
\end{align*}
How do they get to that inequlity?
Best Answer
Let $\phi(x) = x^2$, $(a,b) = (-2,2)$ and $[c,d] = [-1,1]$. At $x = \frac{1}{2}$ we have: $$ \lim_{h\to 0^+} \frac{\phi(x+h) - \phi(x)}{h} = \phi'(x) = 1 > 0 = \frac{\phi(d)-\phi(c)}{d-c} $$