If $ a,b,c,d,e$ are positive numbers bounded by $ p$ and $ q$, i.e, if they lie in $ [p,q], 0 < p$, prove that \[ (a + b + c + d + e)\left(\frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} + \frac {1}{e}\right) \le 25 + 6\left(\sqrt {\frac {p}{q}} - \sqrt {\frac {q}{p}}\right)^2\] and determine when there is equality.