Consider a fixed upfront investment at $t_0=0$ and wlog assume the cashflow to be $-1$. And assume the subsequent returns on this investment are all positive, i.e., a series of positive cashflows $c_{i}>0$ that happens at $t_i > t_0$ where $0=t_0<t_1<\cdots<t_n<\infty$. Now treat all $t_i$ as fixed, but treat $c_i>0$ as variables. Then IRR $r$ corresponding to $(c_1,\cdots,c_n)$ can be defined by $$\sum_i \frac{c_i}{(1+r)^{t_i}}=1$$ So $r$ is an implicit function in $(c_1,\cdots,c_n)\in\Bbb R_n^+$. Is there any guarantee on the convexity/concavity of $r$ under the following assumptions respecitively:
1). $0<t_1<\cdots<t_n\le 1$.
2). $1 <t_1<\cdots<t_n<\infty$.
3). No further assumptions on $t_i$.
From 1D heuristics I believe $r$ is convex under 1) and concave under 2) and indetermined under 3). Are there any easy ways to prove these guesses?
The comment by @Raskolnikov sugggests that these conditions are probably not enough to characterise convexity/concavity.
It seems even the one dimensional case is not as clear cut:
Take $1<t_1<t_2<t_3$, $c_1=c_2=c_3=c$ and thus
$$\frac{1}{\left(1+r\right)^{t_1}}+\frac{1}{\left(1+r\right)^{t_2}} + \frac{1}{\left(1+r\right)^{t_3}}=\frac{1}{c} \; .$$
One can then find relationships $r(c)$ that are neither convex nor concave for appropriate choices of the times. Here's an example I have constructed with the help of the Desmos calculator:
Also, this may be a trivial remark, but the fact you order the times is irrelevant, because the terms in the sum are interchangeable anyway.