Quasiconvex and Quasiconcave functions

Let $f:S ightarrow mathbb{R}$ where $S subset mathbb{R}^n$ is a non-empty convex set. The function f is said to be quasiconvex if for each $x_1,x_2 in S$, we have $fleft ( lambda x_1+left ( 1-lambda ight )x_2 ight )leq maxleft { fleft ( x_1 ight ),fleft ( x_2 ight ) ight },lambda in left ( 0, 1 ight )$

For example, $fleft ( x ight )=x^{3}$

Let $f:S ightarrow R $ where $Ssubset mathbb{R}^n$ is a non-empty convex set. The function f is said to be quasiconvex if for each $x_1, x_2 in S$, we have $fleft ( lambda x_1+left ( 1-lambda ight )x_2 ight )geq minleft { fleft ( x_1 ight ),fleft ( x_2 ight ) ight }, lambda in left ( 0, 1 ight )$

Remarks

Every convex function is quasiconvex but the converse is not true.

A function which is both quasiconvex and quasiconcave is called quasimonotone.

Theorem

Let $f:S ightarrow mathbb{R}$ and S is a non empty convex set in $mathbb{R}^n$. The function f is quasiconvex if and only if $S_{alpha} =left ( x in S:fleft ( x ight )leq alpha ight }$ is convex for each real number alpha$

Proof

Let f is quasiconvex on S.

Let $x_1,x_2 in S_{alpha}$ therefore $x_1,x_2 in S$ and $max left { fleft ( x_1 ight ),fleft ( x_2 ight ) ight }leq alpha$

Let $lambda in left (0, 1 ight )$ and let $x=lambda x_1+left ( 1-lambda ight )x_2leq max left { fleft ( x_1 ight ),fleft ( x_2 ight ) ight }Rightarrow x in S$

Thus, $fleft ( lambda x_1+left ( 1-lambda ight )x_2 ight )leq maxleft { fleft ( x_1 ight ), fleft ( x_2 ight ) ight }leq alpha$

Therefore, $S_{alpha}$ is convex.

Converse

Let $S_{alpha}$ is convex for each $alpha$

$x_1,x_2 in S, lambda in left ( 0,1 ight )$

$x=lambda x_1+left ( 1-lambda ight )x_2$

Let $x=lambda x_1+left ( 1-lambda ight )x_2$

For $x_1, x_2 in S_{alpha}, alpha= max left { fleft ( x_1 ight ), fleft ( x_2 ight ) ight }$

$Rightarrow lambda x_1+left (1-lambda ight )x_2 in S_{alpha}$

$Rightarrow f left (lambda x_1+left (1-lambda ight )x_2 ight )leq alpha$

Hence proved.

Theorem

Let $f:S ightarrow mathbb{R}$ and S is a non empty convex set in $mathbb{R}^n$. The function f is quasiconcave if and only if $S_{alpha} =left { x in S:fleft ( x ight )geq alpha ight }$ is convex for each real number $alpha$.

Theorem

Let $f:S ightarrow mathbb{R}$ and S is a non empty convex set in $mathbb{R}^n$. The function f is quasimonotone if and only if $S_{alpha} =left { x in S:fleft ( x ight )= alpha ight }$ is convex for each real number $alpha$.