Convergence of a sequence of real convex analytic functions

Question

This is a question on the convergence of a sequence of real, convex, analytic functions (it does not get better than that!):

Let $(f_n)_{n\in \mathbb N}$ be a sequence of convex analytic functions on $\mathbb R$.

Suppose that $f_n(x) \to f(x)$ as $n \to \infty$ for all $x \in \mathbb R$ (or in $\mathbb R^+$).

Is $f(x)$ analytic?

Answer 1

Counterexample: Define $f_n(x) = (x^2+1/n)^{1/2}.$ Then each $f_n$ is analytic and convex on $\mathbb R.$ Clearly $f_n(x)\to |x|$ pointwise everywhere. (A little more work shows $f_n(x)\to |x|$ uniformly on $\mathbb R.$)

Answer 2

No—not even necessarily differentiable! The function $f_n(x) = \frac1n\log(1+e^{nx})$ is convex and analytic on $\Bbb R$, but $$ \lim_{n\to\infty} \frac1n\log(1+e^{nx}) = \begin{cases} 0, &\text{if } x\le 0, \\ x, &\text{if } x\ge0. \end{cases} $$