Is there a name for this inequality?

Yes this is the HM-AM inequality and it is a consequence of AM-GM.

Take a look here

RMS-AM-GM-HM inequalities


This is just saying that the inverse of the mean is at most the mean of the inverses:

$$\frac{1}{\frac{a+b}2} \leq \frac{\frac{1}{a} + \frac{1}{b}} 2$$

This follows from the convexity of the function $\frac 1 x$ (on the positive real numbers). Of course, convexity is closely related to the AM-GM inequality, so maybe it's not surprising that this can also been seen as related to AM-GM.

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