A relationship on real functions

Given two functions f, g :  ℝ → ℝ, we say that f is strongly greater than g, denoted f ≫ g, iff there exists a nondecreasing function h such that f(x) > h(x) > g(x) for all x in ℝ. ≫ is obviously a strict order on ℝ → ℝ. The following are additional elementary facts about this relationship.

Proposition. Let f, g, f', g', f₊, g_-, h, r : ℝ → ℝ such that f ≫ g, f' ≫ g', f₊(x) ≥ f(x) and g(x) ≥ g_-(x) for all x, h is strictly increasing and r is nondecreasing, and let k be a positive real number. We have

• (f + f') ≫ (g + g'),
• f₊ ≫ g_-,
• h ∘ f ≫ h ∘ g,
• k(f + r) ≫ k(g + r).

For any function f we introduce the following associated functions f_min, f_max:

f_min(x) := min{f(y) : y ≥ x},
f_max(x) := max{f(y) : y ≤ x}.

provided the expressions actually exist for all x in ℝ. These auxiliary functions give us an alternative way of defining ≫.

Proposition. f ≫ g iff f_min and g_max exist and f_min(x) > g_max(x) for all x in ℝ.

Proof. If f ≫ g then there is a nondecreasing function h such that

min{f(y) : y ≥ x} > min{h(y) : y ≥ x} = h(x),
max{g(y) : y ≤ x} < max{h(y) : y ≤ x} = h(x),

hence f_min and g_max are well defined and  f_min(x) > g_max(x) for all x. Conversely if f_min(x) > g_max(x), let us take

h(x) := (f_min(x) + g_max(x))/2;

f_min and g_max are obviously nondecreasing, thus h is also nondecreasing, and for all x in ℝ

 f(x) ≥ f_min(x) > h(x) > g_max(x) ≥ g(x).

As a corollary, if f ≫ g then lim_x→+∞f(x) is lower bounded and lim_x→-∞g(x) is upper bounded.