We denote the $k$th prime number as $p_k$, and if it is required in your calculations let $g_k=p_{k+1}-p_k$ denoting the gaps between primes.
For integers $n\geq 1$, I define the Andrica's function as the arithmetic function (I don't know if this definition and next claim was in the literature; at first isn't related to Andrica's conjecture but I was inspired in it, see this MathWorld) $$\mathcal{A}(n)=\sqrt[n]{\prod_{k=1}^n\left(\sqrt{p_{k+1}}-\sqrt{p_{k}}\right)}.\tag{1}$$
Claim. Then $$\lim_{n\to\infty}\mathcal{A}(n)=0,$$ and thus $$\mathcal{A}(n)=o(1),$$ as a consequence of a combination of the AM-GM inequality with the prime number theorem and the squeeze theorem.
On the other hand I define the Firoozbakht's function as the function defined for integers $n\geq 1$ $$\mathcal{F}(n)=\sqrt[n]{\prod_{k=1}^n p_{k}\left(p_k^{1/k}-1\right)}.\tag{2}$$ (This definition is inspired in Firoozbakht's conjecture, see this Wikipedia.)
Question.
A) (Optional) Can you improve my knowledge of the function defined in $(1)$ with regard to its asymptotic behaviour as $n\to\infty$? Then provide me a better statement than my Claim about the asymptotic behaviour of $\mathcal{A}(n)$ as $n\to\infty$? You can add your statement as a big/little oh statement or as an asymptotic equivalence. If it is a well-known exercise, then refers the literature and I try to search and read those calculations.
B) Hint or details to get a statement about the asymptotic behaviour of $\mathcal{F}(n)$ as $n\to\infty$. Many thanks.
It will be welcome also if you want to add to your calculations some conjecture or conditional statement (on assumption of well-known conjectures), or well numeric computations or graphics for $1\leq n\leq N$ involving our functions $\mathcal{A}(n)$ and $\mathcal{F(n)}$ for a large $N$.
We have (by Theorem 5 in arXiv:1506.03042) $$ p_k^{1+1/k} - p_k = \log^2 p_k - \log p_k - 1 + o(1). $$ Therefore $$ {\cal F}(n) = O(\log^2 p_n) = O(\log^2 n). $$