concave function with sublinear growth

Question

Does there exist a concave, increasing function $h\colon[0,\infty)\to\mathbb{R}$ such that

1. $\lim_{x\to\infty} h(x)=\infty$

2. $\lim_{x\to\infty} h(x)/x=0$

3. There exist sequences of positive numbers $a_n,b_n,c_n,d_n$ which converge to infinity such that:

3a. $\infty>\lim_{n\to\infty} a_n/b_n=\lim_{n\to\infty} c_n/d_n>0$ but

3b. $\lim_{n\to\infty} h(a_n)/h(b_n) \neq \lim_{n\to\infty} h(c_n)/h(d_n)$

Is it possible?

Answer 1

The negation of your property 3. is called regular variation.

Here is an example of a concave not regularly varying function (taken from this paper by Iksanov and Rösler, p.10) :

Take $$f(x) = 2^{-k} x + 2^{k+1} - 3, \;\;\;\; x \in [4^k,4^{k+1})$$

Then for $x_n=4^n$, $y_n=3\cdot4^n$, $$\lim_n \frac{f(2x_n)}{f(x_n)} = 2 \neq \frac{7}{5} = \lim_n \frac{f(2y_n)}{f(y_n)}.$$