Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$

So far I have not been able to prove it, though it seems like an easy statement. Have you seen something like this?

EDIT:

Now that the original claim is proved, we can ask: what is the best possible constant $C$ so that $\lambda-\mu \leq \frac{k}{C}$?