# Tag Info

### If $x,y,\alpha>0$ and $x>y$, then $x^\alpha>y^\alpha$

Hint: Let $\displaystyle k = \frac{x}{y} \implies k > 1.$ How does $(ky)^\alpha$ compare with $y^\alpha$?
• 21.6k
### Prove the constancy of a harmonic function with $\lim_{\vert x\vert\rightarrow\infty}\frac{\vert f(x)\vert}{\ln\vert x\vert}=0$.
The assumption can be weakened to $$\limsup_{|z|\to \infty}{f(x,y)\over \ln|z|}=:r_0<\infty, \ z=x+iy\qquad(*)$$ Fix $r>r_0.$ By $(*)$ we obtain $$f(x,y)-r\ln|z|< 0,\qquad |z|> R$$ for ...