The area underneath a rainbow ? (Picture)

The darker area between the two rainbows of a double rainbow is called Alexander’s band. I can’t say for sure that there is no term for the inner part of a rainbow, but that wikipedia entry doesn’t mention any. So, if such a term does exist, it is safe to say that few readers would recognise it.

In astronomy, the bright edge of a circular body is called a limb, and I think that would be an appropriate word to use for something seen in the sky, like a rainbow. Strictly, we’re not talking about the outer edge of the rainbow (which would be the colurful part), but I think this would be appropriate in casual usage because it is similar to how people talk about the appearance of the Earth’s horizon from orbit. The glowing curve of the atmosphere is called a “limb”, but so is the bright edge of the Earth just inside it.

If you said “the limb of the rainbow is brighter than the surrounding sky”, I think it would be clear what you meant to people familiar with the astronomical term. But this may be one of those cases where the clearest wording is not the most concise wording.

The darker area outside the primary bow is, as mentioned by bobtato,

Alexanders Dark Band, (see top left of picture)

however the area inside the primary bow is composed of other supernumeraries getting smaller and smaller over a few degrees. It is the combined reflections inside the primary bow that make them appear brighter. Actually in reality each of your eyes is seeing multiple different rainbows, and the colours your brain adds up are not seen as exactly the same colour by each eye (just as the person next to you will see them in different positions and intensities). Famously Newton was colour blind and what he saw as blue we could call cyan and his indigo is more commonly classed as blue hence it may be difficult for some people to agree there are two other significant colours. Without Newtons addition of orange the primary scale should have been.

Red, Yellow, Green, Cyan, Blue and a Violet (not Magenta which is red and blue)
However Newtons theory's are considered his laws of color.

To answer the question

The supernumeraries are brighter than Alexander's Dark Band above the rainbow.