Let positive reals $\{a_i\}$, where not all $a_i$ are equal. Does $$ f(\{a_i\}) = (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 $$ hold for $n \le 16$? It is understood that $a_{n+1} = a_1$.
The restriction to $n \le 16$ comes from a known counterexample for $n = 17$ as follows: Arrange the $a_i$ in blocks such that 9 consecutive $a_i =1.1$ and the following 8 consecutive $a_i =1$. Then
$$ f(\{a_i\}) = (9\cdot 1.1^{1.5} + 8)^2 - (9\cdot 1.1 + 8)(8\cdot 1.1^2 + 7 + 2 \cdot 1.1) = -0.0097 <0 $$
Let me further sketch my thoughts why it may be believed that the inequality indeed holds for $n \le 16$.
First, it can be shown that all block arrangements for $n \le 16$ satisfy the inequality, which can be seen as follows. Let a block of $n\cdot p$ many $a_i =A$ and the remaining $n\cdot (1-p)$ many $a_i =B$, where $0<p<1$. Let $B = x^2 A$. Then we have
$$ \frac{1}{A^3}f(\{a_i\}) = (n p + n (1-p) \cdot x^3 )^2 - (np + n (1-p) x^2)(n p -1 + (n (1-p)-1) x^4 + 2 x^2) $$ which can be rearranged to
$$ \frac{1}{n A^3} f(\{a_i\}) = - n p(1 - p) \cdot x^2 (x - 1)^2 +(p + x^2(1 - p))\cdot(x^2 - 1)^2 $$
Since the leading term with $n$ is always negative and the remaining term is always positive, it is clear that for large $n$, $f <0$, whereas for small $n$, $f >0$. Regarding $n=n_0$ for a moment as a real variable, we can determine the $n_0$ where $f=0$ as
$$ n_0 = n_0(p,x) = \frac{(p + x^2(1 - p))\cdot(x^2 - 1)^2}{p(1 - p) \cdot x^2 (x - 1)^2 } $$ Detailed analysis (solving for the minimum) of this function shows that the smallest value will be attained for $p=0.5$ and $x\to 1$. $x=1$ is not allowed since then all $\{a_i\}$ would equal. Now we have $$ \lim_{x\to 1} n_0(p=0.5,x) = 16 $$ Since $x \ne 1$, we always have $n_0(p,x) > 16$, which shows that all block arrangements with $n \le 16$ satisfy the inequality.
Now if it can be shown that indeed block arrangements constitute the "worst case" of the given inequality, the proof would be complete. I didn't manage to do that, though, or to devise other proof or disproof methods.
Note: the problem originated from this question where it was falsely believed that the inequality would hold for all $n$.