Problem: Given an monotonically increasing function $f(x):[0,a]\rightarrow \mathrm{R}^{+}$, approximate $f(x)$ with polynomial function $g_k(x)=\sum_{i=0}^k a_ix^i$ such that $|f(x)-g_k(x)|\leq \epsilon(k) f(x)$. Here the domain is bounded by $a$, a constant and $\epsilon(k)$ is the multiplicative error bound, as a function of $k$. Also note that $f(x)$ is positive in its domain.
Has this problem been studied. I could not find any reference which talks about multiplicative error bounds. All the references, that I found out, talks about guarantees like $|f(x)-g_k(x)|\leq \epsilon(k)$, for a $k$ order polynomial approximation.