Show that $ax+ by+ cz\geq a^{x}.b^{y}.c^{z}$

Question

Let $a, b, c, x, y, z> 0$ and $x+ y+ z= 1$. Show that $$ax+ by+ cz\geq a^{x}.b^{y}.c^{z}$$ I have found the generalization of this inequality: For 2 sets of numbers $\left \{ a_{i} \right \}, \left \{ b_{i} \right \}> 0$ and $\sum_{1}^{n}b_{i}= 1$, we have: $$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq \coprod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$$ I try use Jensen inequality and Weight AM_ GM but I can' t because the LHS is a product with exponent numbers not integer. I hope to the solution. Thanks!

Answer 1

I don't know what Vasc inequalities are, but it results from Jensen's inequality:

Simply compare the logs: as the natural logarithm is a concave function, one has $$\ln\bigl(a^x b^y c^z)=x\ln a+y\ln b+z\ln c\le \ln(ax+by+cz) $$ for all non-negative $x,y,z$ such that $x+y+z=1$.