If a is any positive number except $1$ , and $ x, y, z,$ are REALS no two of which are equal, then
$a^{x}\left( y-z\right) +a^{y}\left( z-x\right) +a^{z}\left( x-y\right) >0$.
It is quite easy to prove the case when $x, y, z$ are RATIONALS:
assuming WLOG that $x>y>z,$ and considering $ x, y, z $ as integers, we apply the AM-GM inequality to the set of numbers $ a^{x}, a^{x}, a^{x},…, a^{z}, a^{z}, a^{z}, …,$ where $a^{x}$ occurs$ \left( y-z\right)$ times and $a^{z}$ occurs $\left( x-y\right)$ times; then, considering $ x, y, z $ as fractions, and denoting them by $p/d, q/d, r/d$ where $p, q, r, d $ are integers and $d$ is positive, we can easily deduce this case from the preceding.
But how to prove this inequality in the real case (and not using the calculus methods)?
