It is known that the merit of a prime gap can be arbitary large : The merit is defined by $$\frac{p_{n+1}-p_n}{\ln(p_n)}$$ , where $p_n$ and $p_{n+1}$ are consecutive primes.
Does a constant $C$ exist such that for every prime $p_n$ the inequality $$p_{n+1}-p_n\le C\cdot \ln^2(p_n)$$ holds, when $p_n$ and $p_{n+1}$ are consecutive primes ? If yes, what is the best known constant $C$ ?