Sub and Super

In this blog post, we define something stronger than superadditivity and subadditivity, then we will make it to its strongest form.

Definitions

Define ${\displaystyle f\left(x\right)}$to be a univariate function that will analyze later.

Likewise, define ${\displaystyle g\left(x,y\right)}$to be a bivariate function.

Then ${\displaystyle f\left(x\right)}$can be said to be Super-g if the following relation holds for all ${\displaystyle x}$and ${\displaystyle y}$in the domain of ${\displaystyle f\left(x\right)}$:

$${\displaystyle f\left(g\left(x,y\right)\right)>g\left(f\left(x\right),f\left(y\right)\right)}$$Similarly, ${\displaystyle f\left(x\right)}$can be said to be Sub-g if the following relation holds for all ${\displaystyle x}$and ${\displaystyle y}$in the domain of ${\displaystyle f\left(x\right)}$:

$${\displaystyle f\left(g\left(x,y\right)\right)<g\left(f\left(x\right),f\left(y\right)\right)}$$However, there exists a third class of functions, namely this:

$${\displaystyle f\left(g\left(x,y\right)\right)=g\left(f\left(x\right),f\left(y\right)\right)}$$We will call the third class of functions Perfect-g.

We define the overall Super-ness constant to be the following improper double integral:

${\displaystyle C=\lim_{L \to \infty}\int_{-L}^{L}\int_{-L}^{L}\left(f\left(g\left(x,y\right)\right)-g\left(f\left(x\right),f\left(y\right)\right)\right)dxdy}$.

The Super-ness constant can be used to perform testing on whether ${\displaystyle f\left(x\right)}$ is Super-g (diverges to ${\displaystyle \infty}$), Sub-g (diverges to ${\displaystyle -\infty }$), or Perfect-g (converges to some real number).

Special Cases

Superadditivity and Subadditivity

${\displaystyle g\left(x,y\right)=x+y}$

Supersubtractivity and Subsubtractivity

${\displaystyle g\left(x,y\right)=x-y}$

Supermultiplicativity and Submultiplicativity

${\displaystyle g\left(x,y\right)=xy}$

Superdivisivity and Subdivisivity

${\displaystyle g\left(x,y\right)=\frac{x}{y}}$

Superexponentiativity and Subexponentiativity

${\displaystyle g\left(x,y\right)=x^{y}}$

Stronger Form of Sub, Super, and Perfect

Stronger Super$${\displaystyle f\left(g\left(x_1, \ldots, x_n\right)\right) > g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right) }$$Stronger Sub$${\displaystyle f\left(g\left(x_1, \ldots, x_n\right)\right) < g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right) }$$Stronger Perfect$${\displaystyle f\left(g\left(x_1, \ldots, x_n\right)\right) = g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right) }$$Stronger Super-Ness Constant

${\displaystyle C=\lim_{L \to \infty}\int_{-L}^{L}\ldots\int_{-L}^{L}f\left(g\left(x_1, \ldots, x_n\right)\right) - g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right)dx_1, \ldots, dx_n}$

Strongest Form of Sub, Super, Perfect

For all ${\displaystyle n\ge 2}$,

Strongest Super$${\displaystyle f\left(g\left(x_1, \ldots, x_n\right)\right) > g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right) }$$Strongest Sub$${\displaystyle f\left(g\left(x_1, \ldots, x_n\right)\right) < g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right) }$$Strongest Perfect$${\displaystyle f\left(g\left(x_1, \ldots, x_n\right)\right) = g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right) }$$Strongest Super-Ness Constant

${\displaystyle C=\lim_{L \to \infty}\int_{-L}^{L}\ldots\int_{-L}^{L}f\left(g\left(x_1, \ldots, x_n\right)\right) - g\left(f\left(x_1\right) \ldots, f\left(x_n\right)\right)dx_1, \ldots, dx_n}$