• arg: argument
  • inf: infimum
  • sup: supremum

arg inf refers to the argument of the function such that the function attains its infimum (or minimum). For example, let $f(x)=x^2-x$. Then arg inf $f(x)=\frac{1}{2}$ because $f(x)$ attains its minimum at $x=\frac{1}{2}$.

When $f$ is a function on a set $A$, the notation: arg $\max\limits_{x \in A}f(x)$ donates the set of elements of $A$ for which $f$ attains its maximum value. This set may be empty, for example, if $f(x)=x$ and $A=(0,1)$, then $f$ has no maximum on $A$, so: arg $\max\limits_{x \in (0,1)} f(x)= \emptyset$. However, $f$ has a supermum: arg $\sup\limits_{x \in (0,1)} f(x)= {1}$.

Paolo.dL asked me about the difference between arg sup/inf and arg max/min, but I was not able to give a complete answer. Help appreciated! Rinconsoleao (talk) 11:10, 19 July 2011 (UTC)

The min is the smallest value in a set. The inf is the greatest lower bound on the set. Frequently the two are the same, but in tricky cases there is an inf even if the min fails to exist. The relation between max and sup is analogous. Examples:

minimize $y=x^2+5$ by choosing $x$ in $-1 \leq x \leq 1$. In this case the min is $y=5$ at the arg min $x=0$. Also the arg inf $x=0$.

minimize $y=x^2+5$ by choosing $x$ in $1 < x \leq 2$. In this case the min and tthe arg min do not exit because you would like to choose $x=1$, but that’s not in the choice set. In this case the inf is 6, which is the largest number less than or equal to $x^2+5$ for all $x$ in $1<x \leq 2$.

Unfortunately in this second example, I’m not sure whether it’s technically correct to say that the arg inf is 1, or that the arg inf is undefined. Rinconsoleao (talk) 10:57, 19 July 2011 (UTC)

Summarizing: arg inf and arg sup are not synonyms for arg min and arg max, but in the cases when they fail to be equivalent I am not sure whether arg inf and arg sup are well-defined. Help appreciated. Rinconsoleao (talk) 11:07, 19 July 2011 (UTC)

Probably we need a separate article Argument (mathematics) to sort this out. According to MathWorld, “An argument of a function $f(x_1, \dots, x_n)$ is one of the $n$ parameters on which the function’s value depends.” Since in your example there is only one argument $x$, I guess arg inf $f(x)$, where $f(x)=y$, is well defined.