Going through some old papers, I came up with a simple-looking problem I thought about 5 years ago or so.

MO wants motivation ... Associated to a probability measure on a metric space is something called "quantization dimension" ... this involves defining a function $D \colon (0,\infty) \to (0,\infty)$. Exactly how is not the point here, but see for example

http://www.ams.org/mathscinet-getitem?mr=1877974

Lindsay, L. J. and Mauldin, R. D. Quantization dimension for conformal iterated function systems. Nonlinearity 15 (2002), no. 1, 189--199.

It was observed numerically that $D$ is increasing and concave, but proof was lacking. When we do this for the simplest possible self-similar measure (similarities with ratios $s_1, s_2$ and probabilities $p_1, p_2$) I still did not solve it, even though it looks like an elementary calculus exercise. Here it is.

Let $s_1, s_2, p_1, p_2$ be positive real numbers such that $s_1 < 1$, $s_2 < 1$, $p_1+p_2=1$. For $r>0$ define $D = D(r)$ implicitly by $$ \left(p_1 s_1^r\right)^{D/(r+D)} + \left(p_2 s_2^r\right)^{D/(r+D)} = 1. $$ Then:
Does it follow that $D'(r) \ge 0$? [YES]
Does it follow that $D''(r) \le 0$? [OPEN]

At least it was open back then!