In my recent researches, I encountered functions $f$ satisfying the following functional inequality:


$$
(\*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}.
$$


Since $f$ is convex (because $\displaystyle f(x)=\sup\_y [f(y)+f(y)(x-y)]$), it is left and right differentiable. Also, it is obvious that all functions of the form $f(t)=ce^t$ with $c\geq 0$ satisfy
$(\*)$. Now, my questions:


(1) Is $f$ everywhere differentiable?


(2) Are there any other solutions for $(\*)$?


**(3)** Is this functional inequality **well-known** (any **references**
(paper, book, website, etc.) for such functional inequalities)?


Thanks in advance



Are such functions differentiable?
Replace $x$ with $x+y$ to get
$f(x+y)\ge f(y)(1+x)$ or $f(x+y)-f(y)\ge xf(y)$.
Replace $y$ with $x+y$ and then interchange $x$ and $y$ to get $f(x+y)-f(y)\le xf(x+y)$.
Together,
$$
xf(y)\le f(x+y)-f(y)\le xf(x+y).
$$
Dividing by $x$ and taking the limit as $x\to0$ implies that $f$ is differentiable with $f'=f$.


