J, 37 bytes

(34 as an expression without the assignment to variable f)

f=:1{(m&.:^.,m&.:%,m&.:*:,m=:+/%#)^:_

Try it online!

1{ Take the element #1 of
( the function
 m&.:^. {do "m" under natural logarithm}
 ,m&.:% {do "m" under inversion}
 ,m&.:*: {do "m" under square}
 ,m=:+/%# {do "m"} - where m = arithmetic mean
)^:_ applied infinitely many times

For functions a and b, a &.: b ("a under b" (related challenge)) is equivalent to (b inv) a b -- apply b, then a, then inverse of b. In this case, geometric/harmonic/quadratic mean is the arithmetic mean "under" logarithm, inversion, and square respectively.

J, 34 bytes

(31 as an expression without the assignment to variable f)

f=:1{(^.z,%z,*:z,[z=:(+/%#)&.:)^:_

Try it online!

For functions a and b, a &.: b ("a under b" (related challenge)) is equivalent to (b inv) a b -- apply b, then a, then inverse of b. In this case, geometric/harmonic/quadratic mean is the arithmetic mean "under" logarithm, inversion, and square respectively.

J, 37 bytes

(34 as an expression without the assignment to variable f)

f=:1{(m&.:^.,m&.:%,m&.:*:,m=:+/%#)^:_

Try it online!

J, 37 bytes

(34 as an expression without the assignment to variable f)

f=:1{(m&.:^.,m&.:%,m&.:*:,m=:+/%#)^:_

Try it online!

1{ Take the element #1 of
( the function
 m&.:^. {do "m" under natural logarithm}
 ,m&.:% {do "m" under inversion}
 ,m&.:*: {do "m" under square}
 ,m=:+/%# {do "m"} - where m = arithmetic mean
)^:_ applied infinitely many times

For functions a and b, a &.: b ("a under b" (related challenge )) is equivalent to (b inv) a b -- apply b, then a, then inverse of b. In this case, geometric/harmonic/quadratic mean is the arithmetic mean "under" logarithm, inversion, and square respectively.