Sum of monotone functions

Question

Assuming $\, f_i(x), i=1,...,N\,$ are real monotone (increasing or decreasing) functions, what can we say about $g(x)=\sum_i f_i(x)$? Does this extra information gives us more insight about $g(x)$?

Answer 1

By induction on $N \ge 1$, for any reals $a_1, \dots, a_N, b_1, \dots, b_N$ with $a_i < b_i$ for all $i = 1, \dots, N$, we have: $$ \sum_{i=1}^N a_i < \sum_{i=1}^N b_i \text{.} $$ Assume first that the $f_i$ are all monotone increasing (and that this means strictly). In any case we assume that they're all "the same kind of monotone".

Given reals $x, y$ with $x < y$, letting $a_i = f_i(x)$ and $b_i = f_i(y)$, we have $a_i < b_i$ for all $i$, so: $$ g(x) = \sum_{i=1}^N a_i < \sum_{i=1}^N b_i = g(y)\text{,}\tag{*} $$ so $g$ is monotone increasing too.

Similarly if the $f_i$ are monotone decreasing (replace "$<$" with "$>$" in (*)), or if they're monotone "nondecreasing" (replace "$<$" with "$\le$") or monotone "nonincreasing".

A simple counterexample shows that the sum of monotone functions of different kinds isn't necessarily monotone: let $f_1$ be the identity on $[0,1]$ and constantly 1 on $[1,2]$; let $f_2$ be constantly $0$ on $[0,1]$ and $x \mapsto 1-x$ on $[1,2]$. Then $f_1$ (resp. $f_2$) is monotone nondecreasing (resp. nonincreasing), but $g = f_1 + f_2$ is no kind of monotone — it's a half-cycle of a triangle wave.

Answer 2

Firstly, I begin with a very simple observation: if $f_1 , \dots , f_N$ are all increasing functions, then their sum is increasing. In fact, for all $x < y$, you have $f_i(x) \le f_i(y)$, so summing up you get $$f_1(x) + \dots +f_n(x) \le f_1(y) + \dots +f_n(y)$$ i.e. $\sum_i f_i$ is increasing. Moreover if one of them is strictly increasing, then the sum is strictly increasing (you have $<$ instead of $\le$).

Now, in the general case, you have that some of the $f_i$s are increasing, while the other ones are decreasing. Summing together increasing ones you get an increasing function $F$. Summing together decreasing ones you get a decreasing function $G$. Note that $-G$ is an increasing function, so that you have $$\sum_i f_i = \sum_{\mbox{increasing}} f_i + \sum_{\mbox{decreasing}} f_i = F+G= F-(-G)$$ so that $\sum_i f_i$ is a difference of two increasing functions. And now, about monotonicity of $\sum_i f_i$ nothing can be said. For example, if you take $F(x)=x^3$ and $G(x)=-x$, you have $F(x)+G(x)=x^3-x = x(x-1)(x+1)$ which is not monotone since it has three distinct zeroes.