Using a simple java application, I have noticed that for $x > 25$:

$$\psi\left(\frac{x}{5}\right) \ge \psi\left(\frac{x}{3}\right) - \psi\left(\frac{x}{4}\right)$$

where:

$$\psi\left(x\right) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$

and

$$\vartheta\left(x\right) = \sum_{p \le x} \log p$$

As I understand it, based on the Prime Number Theorem, this inequality should be true.

Is this inequality elementary and straight forward to prove? Or is this more difficult to prove than it appears?

Thanks very much,

-Larry