Let $(a_1, a_2, \dots) \in \mathbb{R}^\infty$ be a fixed sequence of real constants, and suppose the rule $$ x \mapsto \sum_{n = 1}^\infty a_n x^n $$ defines a function from the nonempty open interval $(-b, b)$ ($b > 0$) to $\mathbb{R}$. Denote this function by $f$.

Define the function $g$ as follows $$ g(x) := \sum_{n = 1}^\infty |a_n| x^n $$ and suppose $g$ is well defined inside $(-b, b)$.

Now consider the functions $F := \exp \circ f$ and $G := \exp \circ g$, and let $(u_0, u_1, \dots)$ and $(v_0, v_1, \dots)$ be their Taylor coefficients, respectively, that is $$ \begin{align} F(y) & = \sum_{n = 0}^\infty u_n y^n \\ G(y) & = \sum_{n = 0}^\infty v_n y^n \end{align} $$

  1. Is there some nonempty interval $(-c, c)$ ($0 < c < b$) inside of which the Taylor expansions of $F$ and $G$ converge to $F$ and to $G$, respectively?
  2. If the Taylor expansions of $F$ and $G$ converge to $F$ and to $G$, respectively, inside some nonempty interval $(-c, c)$ ($0 < c < b$), is it the case that $|u_n| \leq v_n$ for all $n \in \{0, 1, \dots\}$?

The problem has been solved below by user Nathanson. For my future reference, I'd like to follow up on his answer by providing the following reference.

Theorem 104 on p. 180 of Konrad Knopp's "Theory and Application of Infinite Series" (Dover Publications, 1990) states that, denoting by $\alpha$, $\beta$ the two real power series $\alpha(x) = \sum_{n = 0}^\infty a_n x^n$ and $\beta(y) = \sum_{n = 0}^\infty b_n y^n$, the composition $\beta \circ \alpha$ is again a power series $\sum_{n = 0}^\infty c_n x^n$, whose circle of convergence contains (at a minimum) all those $x$'s for which $\sum_{n = 0}^\infty |a_n x^n|$ converges to a number within $\beta$'s interval of convergence. The coefficients $c_n$ are obtained by grouping like powers of $x$ together in the expansion $$ b_0 + b_1 (a_0 + a_1x + \cdots) + b_2 (a_0 + a_1x + \cdots) + \cdots $$ In other words, $$ c_n = \sum_{k = 0}^\infty b_k a_n^{(k)} $$ where $a_n^{(k)}$ is the coefficient of $x^n$ in the power series expansion of $\left(\sum_{n = 0}^\infty a_n x^n\right)^k$. The coefficient $a_n^{(k)}$ is constructed from the coefficients $a_0, a_1, \dots, a_n$ in a perfectly determined manner as a finite sum of finite products of these $n$ coefficients.

  1. The functions $f$ and $g$ viewed as functions over the complex are analyticon a disc with center $0$ and radius $b$. Therefore $F$ and $G$ are also analytic on the same disc. This implies that their Taylor expansions converge on that disc. In particular they converge on the same interval $(-b,b)$.

  2. Yes, use triangle inequality and chain rule (Faa di Bruno's formula).

  • $\begingroup$ Thank you. Are the Taylor coefficients $\{u_n\}_{n=1}^\infty$ of $F$ and $\{v_n\}_{n=1}^\infty$ of $G$ necessarily real? $\endgroup$ – Evan Aad Mar 25 '15 at 15:30
  • $\begingroup$ @EvanAad Yes, they are derivatives of those real functions. $\endgroup$ – Nathanson Mar 25 '15 at 15:44

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