Construct an entire power series

Question

Let $g:\mathbb{R}_+ \to \mathbb{R}_+$ be monotone increasing. Prove that there exists an entire power series $f(x)=\sum a_n x^n$(i.e. with infinite radius of convergence) s.t. $$\forall x>0, f(x)>g(x)$$

I can't find an appropriate function, for I think there's too little info about $g$, anyone has an idea?

Thanks a lot~

Context? Where did this problem arise: in a contest, in a particular math course...? — Greg Martin, May 28 '17 at 02:51
Three upvotes for zero context (and a repeating offender in these matters). Life is nice... — Did, May 28 '17 at 14:55

score 1 · Accepted Answer · answered May 28 '17 at 04:47

Suggestion: Consider

$$g(1) + g(2)x + g(3)\left (\frac{x}{2}\right)^{n_2} + + g(4)\left (\frac{x}{3}\right)^{n_3} + \cdots, $$

where the $n_k$ are chosen so that $1< n_2 < n_3 < \cdots$ and

$$g(k+1)\left (\frac{k-1}{k}\right)^{n_k} < \frac{1}{2^k}$$

for $k\ge 2.$

Construct an entire power series

1 Answers1