PROVE THAT: Data: $a,b,c,x,y,z > 0$ $1/x+1/y+1/z=1$ ?

Question

We must prove:
$abc<a^x/x+b^y/z+c^z/z$

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Answer 1 · 2017-08-28T09:54:45

See below.

With $x_k > 0$ , from $sum_(k=1)^n x_k ge (prod_(k=1)^n x_k)^(1/n)$ we can derive

$mu_1 x_1+mu_2 x_2+mu_3x_3 ge x_1^(mu_1) x_2^(mu_2) x_3^(mu_3)$

with $mu_1+mu_2+mu_3=1$ now choosing

${(x_1=a^x),(x_2=b^y),(x_3=c^z),(mu_1=1/x),(mu_2=1/y),(mu_3=1/z):}$

we get

$a^x/x+b^y/y+c^z/z ge a b c$