TITLE: What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$
QUESTION [5 upvotes]: I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$.
In the paper, he uses the following bounds for the second Chebyshev function $\psi(x)$:
$$1.086x > \psi(x) > 0.916x - 6.954$$
If I apply the better upper bound from Rosser & Schoenfeld, 1962 of:
$$1.03883x > \psi(x)$$
Then Nagura's proof shows that there is always a prime between $x$ and $\frac{8x}{7}$ when $x \ge 34$.
Is this the best upper and lower bound for $\psi(x)$:  
$$1.03883x > \psi(x) > 0.916x - 6.954$$
Does anyone know of any results that improve on these bounds?
Thanks,
-Larry

REPLY [4 votes]: The most recent results on bounds for $\psi(x)$ are from this year:
Sharper estimates for Chebyshev's functions $\vartheta$ and $ψ$, February 2013.

In this article we present some
  improved results for Chebyshev's
  functions $\vartheta$ and $\psi$ using
  the new zero-free region obtained by
  H. Kadiri and the first
  $10^{13}$ zeros of the Riemann zeta
  function on the critical line calculated by
  Xavier Gourdon. The methods in the
  proofs are similar to those of the
  Rosser-Shoenfeld papers on this
  subject.