[Note: Rolled back to previous version, before I edited it. This question was poorly posed and didn't cover the cases I had in mind. See this new question.]
Let—$$f(x) = \sum_k a_k x^k, \text{ and } g(x) = \sum_k b_k x^k,$$ be functions that map [0, 1] to [0, 1]. The functions are infinite convergent power series with non-negative coefficients.
Then is $g(x)$ bounded above by $f(x)$ only if $b_k$ is bounded above by $a_k$ for every $k$? Is this result known?
For example, I note that $\exp(x/4)/2$ and $\exp(x/2)/2$—
- are bounded above by $\sum_k (1/2)(1/2)^k x^k$, which is the probability generating function of a geometric(1/2) random variable, and also
- have coefficients that are bounded above by that generating function,
which is why I suspect this is a more general phenomenon.
No. Take for instance $f(x)=x,\;g(x)=x^2$. Then $g(x)$ is bounded above by $f(x)$, but $b_2=1$ is not bounded above by $a_2=0$.