What is the need of homogeneous function?

Question

I understand what an homogeneous function is, but I just can't seem to understand the purpose of having this concept of function. What does it even tell us? I mean if $f(tx,ty) =t^n f(x,y)$ then what does it tell us? What is the use of this information?

Answer 1

A homogeneous function can be "decomposed" as $\,f(x,y)=y^n \cdot f\left(\frac{x}{y},1\right)= y^n \cdot g\left(\frac{x}{y}\right)\,$ at points $\,y \ne 0\,$, which can provide significant insights in many scenarios.

Answer 2

Knowing the values of the function along the unit circle centered at zero will tell you the value of the function everywhere.

Rewriting in polar coordinates you have $f(r, \theta) = r^n f(1, \theta)$. And along this radial line of angle $\theta$ you know the function grows with power $n$ of the radius.

More generally, knowing the function value at a point lets you know the function value along all points on the ray through the origin and that point.

Answer 3

Homogenous functions are helpful in proving inequalities.

For example, consider the Nesbitt's inequality:

For positive $a,b,c$, prove $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac32$, with equality when $a=b=c$.

See this link for many different proofs, among which there is the one called "normalization" (normalizing the homogenous inequality).

Why the original problem is equivalent to the following:

For positive $a,b,c$ and $a+b+c=1$, prove $\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}\ge \frac92$, with equality when $a=b=c=\frac13$?

Because the function $f(a,b,c)=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$ is homogenous (of degree $0$). Indeed: $$f(ta,tb,tc)=\frac{ta}{tb+tc}+\frac{tb}{tc+ta}+\frac{tc}{ta+tb}=t^0f(a,b,c).$$ Further details: Let $a+b+c=t>0$. If $a=tx, b=ty, c=tz$, then: $x,y,z>0, x+y+z=1$ and the original inequality transforms to: $$\frac{tx}{ty+tz}+\frac{ty}{tz+tx}+\frac{tz}{tx+ty}\ge \frac32 \Rightarrow \\ \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\ge \frac32 \Rightarrow \\ \frac{x}{y+z}+1+\frac{y}{z+x}+1+\frac{z}{x+y}+1\ge \frac32+3 \Rightarrow \\ \frac{x+y+z}{y+z}+\frac{y+z+x}{z+x}+\frac{z+x+y}{x+y}\ge \frac92 \Rightarrow \\ \frac{1}{y+z}+\frac{1}{z+x}+\frac{1}{x+y}\ge \frac92.$$

Answer 4

A useful result in the theory of partial differential is based on such homogeneous function is called Euler's theorem.

If $u=f(x,y)$ is homogeneous of degree $n$ then it always satisfy the partial differential equation of first order $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu$.