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

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

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

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?