05AB1E, 1818 16 bytes
-2 bytes thanks to Erik the Outgolfer
‚©nO;t®O;®Pt2®zOnO;t¹O;¹Pt2¹zO/O
Explanation:
‚© Pair the input a, b into [a, b] and store it in the registry (without popping)
nO;t Root mean square
n Raise [a, b] to [a ** 2, b ** 2]
O Sum
; Half
t Square root
®O;¹O; Arithmetic mean
®¹ Retrieve stored [a, b]
O Sum
; Half
®Pt¹Pt Geometric mean
®¹ Retrieve stored [a, b]
P Product
t Square root
2®zO2¹zO/ Harmonic mean
®¹ Retrieved stored [a, b]
z Vectorised inverse to [1 / a, 1 / b]
O Sum
2 / Get 2 divided by the sum
O Sum of all elements in stack