Trivial lower bounds for single point mass delta-v problems?

Question

Given two arbitrary elliptic orbits around an ideal single point mass, there will always exist a transfer with the minimal $\Delta v$ required.

It's easy to find an upper bound for this ideal transfer:

$$\Delta v \leq \sqrt{\frac{2}{r_{P1}}} - v_{P1} + \sqrt{\frac{2}{r_{P2}}} - v_{P2}$$

This is a general bi-elliptical transfer, and it's sometimes not possible to do better than that.

It sets a maximum $\Delta v$ required for arbitrary transfers. Are there any trivial lower bounds for the minimal $\Delta v$ required?

Answer 1

While it's a very loose lower bound, it does perhaps have some value to present one most trivial such bound.

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With basis in the fact that under no circumstances there exists any more efficient way to increase apoapsis than a prograde burn at periapsis, the following bound exists:

$$\Delta v \geq \sqrt{\frac{2}{r_{P1}} - \frac{2}{r_{P1} + r_{A2}}} - \sqrt{\frac{2}{r_{P1}} - \frac{2}{r_{P1} + r_{A1}}}$$

(a simple vis-viva calculation)

Assuming $r_{A2} \geq r_{A1}$, which can be assumed without any loss of generality since the orbits could otherwise be swapped.

Inclination and argument of periapsis obviously add some non-zero cost on top of this, but it's nevertheless a lower bound.