You talk about increasing gain below threshold. An
upward compressor does this.
You mention "inverse expansion" - this is normally called an
upward expander (sometimes called an "uncompressor"), and works differently than the desired result you decribe.
Here is a description of both.
With an upward expander the loud parts are made louder. Instead of reducing gain above threshold like a normal compressor, it will increase gain. Useful for transient enhancements. You may think of this a bit like a normal compressor with additional make-up gain.
With upward compression the quit parts made louder. This will typically raise the noise floor, so be careful with the settings. It may be useful for raising the general RMS like seen with parallel compression (mixed wet/dry).
Check out this article for additional info on
uplift compression and
Dynamic Functions In General
Here is a simple guide on the various dynamic functions. The gate is not listed - a gate is like downward expander (or simply "expander"), only the output is completely cut off below threshold (the lower vertical line is completely vertical).
The input signal is represented on the x-axis, and the output signal is on the y-axis. A completely untouched signal would be straight linear from lower left corner towards upper right corner (same in..same out):