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