Let $c>0$ be a constant real number.
(a) Is there any non-polynomial real function $f$ satisfying the inequality $$ -\frac{1}{c}x^2+\frac{1}{c}yx<f(y)-f(x)<\frac{1}{c}y^2-\frac{1}{c}xy\; ; \;\;\; x< y ? $$
If yes, are there infinitely many such functions?
(Note that the function $f(t)=\frac{1}{2c}t^2$ is a solution, but it is a polynomial)
(b) Does exist another polynomial solution (except that $f(t)=\frac{1}{2c}t^2$)?
This is equivalent to $x < \frac{c(f(y)-f(x))}{y - x}$ < y (factor $y-x$ out on the outsides and divide by it) so in particular the limits as $y$ tends to $x$ from above and as x tends to y from below give differentiability and $f'(x) = \frac{x}{c}$ at every $x \in \mathbb{R}$. So your solution plus an arbitrary constant is the only kind of solution.