Question

I'm looking for the example of a concave function $g\colon [0,1]\mapsto \mathbb{R}$, with $g(0)=0$, for which

1. $\lim\limits_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ and
2. $\lim\limits_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}=1$ for every $\lambda>1$

Answer 1

Take $g_1(x)=x\log^2x$. Properties 1,2 are evidently satisfied, and computation of the second derivative shows that it is negative for $0< x<1/e$. Now rescale: $g(x)=g_1(x/e)$.