Under the thumb
Completely under someone's control.
Few fields of endeavour have as much false etymology associated with them as falconry. 'A bird in the hand is worth two in the bush' derives from hunting with birds of prey and this has seemingly led to a spate of supposed derivations along the same lines. 'Hoodwinked' is said to derive from putting a cloth cap over the eyes of a falcon to calm the bird by simulating nighttime and 'old codger' is said to derive from the carrier of the cadge or cage that held the birds. Neither of these has any basis in fact. We can add to the list 'under the thumb', which was recently presented on the BBC's Alan Titchmarsh Show as originating from the handler's use of his thumb to trap the bird in his hand when he didn't want it to fly. Again, that's pure flight of fancy.
Being 'under one's thumb' is just a figurative expression that alludes to being completely under another's control. The allusion was to a protagonist so powerful and a victim so insignificant that even the former's thumb was strong enough to control them. None of the early references relates to falconry, or to any other specific origin. The phrase was always used to refer to a powerful person, like a king or lord, exercising control over a subject. The earliest reference I can find to the expression in print is from the 18th century English politician Arthur Maynwaring in The Life and Posthumous Works of Arthur Maynwaring, 1715, which, as is strongly hinted at in the title, was printed after his death:
The French King having them under his Thumb, compell'd them to go at his Pace.
The phrase was well enough established by the 19th century for it to spawn the verb form 'thumbing'. This was defined in James Halliwell's Dictionary Of Archaic And Provincial Words, 1847, as:
Thumbing: A Nottingham phrase, used to describe that species of intimidation practised by masters on their servants when the latter are compelled to vote as their employers please.
'Under the thumb' is still used today with much the same meaning as when it was coined in the 18th century. It's not entirely clear what meaning Mick Jagger and Keith Richards had in mind when they wrote the eponymous 'Under My Thumb' in 1966, but it probably wasn't falconry.