<s>[INST] Inequality with 6 variables I am trying to show the following,
Let, $x,y,z,a,b,c$ be positive reals such that $x+y+z=1$
Then $ax + by + cz +2 \sqrt{(ab+bc+ca)(xy+yz+zx)} \leq a+b+c$
This is practice problem 6 in this document
This is equivalent to 
$\sqrt{(ab+bc+ca)(xy+yz+zx)} \leq \frac{(y+z)a+(x+z)b+(x+y)c}{2(x+y+z)}$
But I am not avle to apply Jensen's inequality. Thank you for help
 [/INST] since Use twice Cauchy-Schwarz inequality,
\begin{align*}
&ax+by+cz+2\sqrt{(ab+bc+ac)(xy+yz+xz)}\\
&\le\sqrt{(a^2+b^2+c^2)(x^2+y^2+z^2)}
+\sqrt{(2ab+2bc+2ac)(2xy+2yz+2xz)}\\
&\le \sqrt{(a^2+b^2+c^2+2ab+2bc+2ac)(x^2+y^2+z^2+2xy+2yz+2xz)}\\
&=(a+b+c)(x+y+z)\\
&=a+b+c
\end{align*}
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