I would like to know if is feasible a nice/potentially interesting combination of the Firoozbakht's conjecture, this Wikipedia, and the Ramanujan tau function using the so-called Ramanujan conjectures (now theorems), see this section of Wikipedia.
My combination using two of those Ramanujan conjectures was the inequality (if there are no typos)
$$\tau(p_n^{n+1}p_{n+1}^n)>(\tau(p_{n+1})\tau(p_{n+1}^{n-1})-p_n^{11(1+1/n)}\tau(p_{n+1}^{n-2}))\cdot(\tau(p_{n})\tau(p_{n}^{n})-p_n^{11}\tau(p_{n}^{n-1}))\tag{1}$$ for $n>2$.
Question. Is it posible to get an inequality with good mathematical content/meaning* combining the Firoozbakht's conjecture and the properties (the previous mentioned Ramanujan's conjectures) of the Ramanujan $\tau(m)$ function? Many thanks.
*I know that the $\tau(m)$ function is erratic and its particular values very large, that I would like to know an elegant and simple (more simple than my attempt and I prefer if it is feasible that your inequality invoke each of those three conditions) combination of the mentioned properties of the Ramanujan tau function and the mentioned Firoozbakht's conjecture about primes.
References:
[1] Farideh Firoozbakht, Conjecture 30. The Firoozbakht Conjecture, The Prime Puzzles & Problems Connection, by Carlos Rivera (22 August 2012).