Ratio of logarithmic primes Any help is appreciated in proving/disproving the following inequality
$$
\frac{\ln{p_{n+1}}}{\ln{p_{n}}} < \frac{n+1}{n}
$$
 A: This is Firoozbakht’s conjecture. According to the link, it has been verified for primes up to $4\times10^{18}$, but is believed to be false, as it contradicts the Cramér–Granville heuristic. 
A: Not so much an answer as adding another level to the same question.
The Firoozbakht's conjecture (1982) is equal to:
$$(p_{n+1})^{n} < (p_n)^{n+1}.$$
Then the natural log is:
$$n \ln(p_{n+1}) < (n+1)\ln(p_n).$$
Now, $$\ln(p_n) \leq \ln(n) + \ln(\ln(n)) + 1\text{, for $n \geq 2$. (*)}$$
With $n = n+1$ we have:
$$\ln(p_{n+1}) \leq \ln(n+1) + \ln(\ln(n+1)) + 1, \text{for $n \geq 2$.}$$
And, because $$p_n \geq n*\ln(n)\text{, for $n \geq 2$;   (**)}$$
the natural log of $p_n$ is: 
$$\ln(n) + \ln(\ln(n)) \leq \ln(p_n), \text{for $n \geq 2$}.$$
With $n = n+1$ we have:
$$\ln(n+1) + \ln(\ln(n+1)) \leq \ln(p_{n+1}), \text{for $n \geq 2$}.$$
So, if
$$n \ln(n+1) + \ln(\ln(n+1)) \leq n \ln(p_{n+1}) < n(\ln(n+1) + \ln(\ln(n+1)) + 1) < (n+1)(\ln(n) + \ln(\ln(n)))  < (n+1)\ln(p_n) < (n+1) \ln(n) + \ln(\ln(n)) + 1$$
holds then the conjecture is true for all terms. If only the outer terms only hold then the primes with largest gap maximal primes will hold true. Primes with smaller prime gaps require sharper bounds on the inner terms.
With the outer terms, dividing by $n(\ln(n) + \ln(\ln(n)))$ we have:
$$\frac{\ln(n+1) + \ln(\ln(n+1))}{\ln(n) + \ln(\ln(n)) + 1} < \frac{n+1}{n} \text{, for $n \geq 2$}.$$
This inequality is true because the left-side increases slower than the right-side.
With the inner terms, dividing by $n(\ln(n) + \ln(\ln(n)))$ we have:
$$\frac{\ln(n+1) + \ln(\ln(n+1)) + 1}{\ln(n) + \ln(\ln(n))} < \frac{n+1}{n} \text{, for $n \geq 2$}.$$
This inequality is false for every $n$ value tested. 
Is there a problem with any of these statements?
References:
$(*)$ Proved in Dusart 2010, ESTIMATES OF SOME FUNCTIONS OVER PRIMES
WITHOUT R.H.,section 4. Useful Bounds)
$(**)$ Used in Dusart 1999, THE k th PRIME IS GREATER THAN k(ln k + ln ln k − 1) FOR k ≥ 2 Lemma 1. p. 413
