The chords with the 4th string root is primarily used for 7th chords. But I would like to start with the triad chord form.
The triad chords with the 4th string root is the same form as that of the 6th string root under the 4th string. since I had explained it with the 3rd fret as the root, here are the chord forms of the F major and F minor chords with the 3rd F note of the 4th string as the root.

The left side is the F major chord and the right side is the F minor chord. This one can be taken for use as a simple F chord when you are just starting out on the guitar and cannot play the F barre chord. These are in the form of abbreviated versions of the 6th string root. Next, we will write the chord form of the Seventh chord.

Here, the root note of the fourth string is held down with the index finger, as is the sixth and fifth string root. The fourth-string root is often used in this sevens chord. It is used when you want to play chords in the upper register, when you don’t want the 1st and 5th degrees to stand out, or when you are playing a chord solo.
It is also used frequently when playing major seventh chords; the form of the F major seventh chord on the fourth string root looks like the following.

Since the 1st, 2nd, and 3rd strings are aligned vertically, it is easy to hold them in barre with the ring finger; sometimes this sevens chord sounds more stylish than using the major sevens chord with the 6th string root. It would be nice to be able to use them differently.