minimum value of $x^{\ln(y)-\ln(z)}+y^{\ln(z)-\ln(x)}+z^{\ln(x)-\ln(y)}$

Question

If $x,y,z>0.$ Then minimum value of

$x^{\ln(y)-\ln(z)}+y^{\ln(z)-\ln(x)}+z^{\ln(x)-\ln(y)}$

what i try

Let $\ln(x)=a,\ln(y)=b.\ln(z)=c$

So $x=e^{a},y=e^{b},z=e^{c}$

How do i solve it help me

Answer 1

Useful fact

$$\large x^{\log y}=y^{\log x}$$

Also refer to another answer of mine here.

Let $u=x^{\ln y}$, $v=y^{\ln z}$ and $w=z^{\ln x}$, then $u,v,w\in \mathbb{R}^+$

\begin{align} f(x,y,z) &= x^{\ln y-\ln z}+y^{\ln z-\ln x}+z^{\ln x-\ln y} \\ &= \frac{u}{w}+\frac{v}{u}+\frac{w}{v} \\ & \ge 3\sqrt[3]{\frac{u}{w} \times \frac{v}{u} \times \frac{w}{v}} \tag{by AM $\ge$ GM} \\ &= 3 \end{align}

Answer 2

After your substitution you get what seems like a more manageable $$ f(a,b,c) = e^{a(b-c)} + e^{b(c-a)} + e^{c(a-b)} $$ and you can now minimize easily using the standard techniques.