# Defining relations between functions in Mathematica

I would like to find a way to implement the following in Mathematica:

Let $$F$$ be the set of real functions $$f_1,f_2,\ldots f_n$$. The function $$\min$$ is defined as follows...

$$\min F=f_i\iff \forall j.\lim_{x\to\infty}[f_i(x)-f_j(x)]\le0$$

Let $$h:\Bbb{R}^2\to\Bbb{R}$$ and $$f,g:\Bbb{R}\to\Bbb{R}$$. The relation $$R_h$$ is defined as follows...

$$R_h(f,g)\iff\exists x,y.\forall z>y. h(\min\{f,g\}(z),x)>\max\{f,g\}(z)$$

For the first of these, I've opted to define a binary operator FuncLess which behaves for functions much the same way as <= for numbers:

FuncLess[f_,g_]:=Limit[f[x]-g[x],x->Infinity]<=0


This seemed more practical to me since the minimum and maximum can be found using a loop, and I don't need to deal with the 'forall'. Still, if there's a more efficient way to do this, I would like to know.

For the second relation, $$R_h$$, I'm not sure what to do. I tried inputting the expression as-is:

FuncR[h_][f_,g_]:=
If[
FuncLess[f,g],
Exists[{x,y},ForAll[z,z>y,h[f[z],x]>g[z]]],
Exists[{x,y},ForAll[z,z>y,h[g[z],x]>f[z]]]
]


but this proved ineffective. I'm not sure how Mathematica evaluates quantifiers, but I imagine that I'm not using them correctly.

Example

In:= f[x_]:=x^2

In:= g[x_]:=x^3

In:= h[x_,y_]:=x^y

In:= FuncLess[f,g]

Out= True


Ideally, I want FuncR[h,f,g] to return True, but it only returns the expression. I've tried using Reduce[FuncR[h,f,g],{x,y,z},Reals], but this returns the message:

Reduce: This system cannot be solved with the methods available to Reduce

• Can you provide examples for f, g and h? – Carl Woll Jan 1 at 20:33

Maybe you can use AsymptoticGreater/AsymptoticLess:

FuncR[h_][f_,g_] := Switch[AsymptoticLess[f[x],g[x],x->Infinity],
True,
AsymptoticGreater[h[f[z],x], g[z],z->Infinity],

False,
AsymptoticGreater[h[g[z],x], f[z],z->Infinity],

_,
Indeterminate
]


f[x_] := x^2