the real analysis book says that
$$f:\mathbb{R}_+ \rightarrow \mathbb{R}_+$$ where $f$ is strictly increasing and concave function. it has the following property
$$f(ax+(1-a)y) \le f(ax) + f((1-a)y)$$
where $a \in [0,1]$.
This property seems wrong. As far as I know, that property is for convex function, not concave function. I do not think the textbook is wrong. Can you please explain it?
The textbook used it to show the function $d(x,y)$ is a matric in $\mathbb{R}$
Best Answer
hmmmmm. I am not sure. The property you mention is not generally a property (in general) of either concave or convex functions, but actually of sub-additive functions. However, I am inclined to believe that the conjecture, or the book may be correct for the following reasoning.
Technically, (on the average, to abuse terminology) a property that would most likely be shared by concave functions over the positive reals, rather then convex functions. Unless its one of these sub-linear or super-linear functions which are some admixture of both (and almost, both,'convex and concave', or super-additive, and sub-additive) over $[0,1]$)
Notice that in $(1)$ the scope of the multiplicative elements $a$ and $(1-a)$ is within $F$ not outside $F$ as in $(2)$ and $(3)$. If one can show that $F$ is sub-additive or that F(tx)>=tF(x) it should be.
$(1)f(ax+(1−a)y)≤f(ax)+f((1−a)y)$
Convex: $f(ax+(1−a)y) ≤a f(x)+(1−a)f(y)$
Concave: $f(ax+(1−a)y)\geq af(x)+(1−a)f(y)$
I agree that $(1)$ definitely Seems wrong for concave $F$ but it actually may not be. Remember that the "above" $(1)$ is a property that function, $F$ 'sub-additive over the positive reals' may have.
Concave functions are sub-additive, over the positive reals, when $f(0) ≥ 0$,see https://en.wikipedia.org/wiki/Concave_function. See pt 10. Although the domain only specifies positive- reals, not-non-negative reals and may not include the $0$. That is the only the issue. It also not a closed domain and range either.
However, given the positive domain, and strict monotonic increasing-ness, it might have the same effect, and the conjecture MIGHT (I stress) be correct. It will also be strictly quasi convex and strictly quasi concave.
If $F$ is concave and,$f(0) ≥ 0$ then $F$ is sub-additive over the positive reals; so the book, is "arguably" correct.
I (stress) arguably, as the the domain, of the function, only specifies positive- reals, not-negative reals. ie including $0$. Maybe strict mono-tonicity, and posi-tivity may help in this case. Notice, that given sub-additivity.