To leave no stone unturned, briefly let us take notice of the right hand (in the right-handed performer).  The right hand must be FREE to manipulate the slide (on the trombone) or finger the valves on valve instruments. Needless to say, the right hand must be totally relaxed and NOT be used to hold or support the instrument in any way.  This is particularly true of the trumpet and trombone.  The only exception being:  when the performer is using a wa-wa mute or plunger mute that requires the left hand manipulation.  On the trumpet, as to the little-finger ring hook, with the thumb of the right hand placed between the first and second valve casings, this little-finger hook and thumb position facilitates the speed and width of the hand vibrato, widely used by most performers.

Even with other types of vibrato (diaphragmatic, breath, jaw, lip, etc.) there must be no excuse to use the right hand to support the instrument and jam the instrument into the face, using the little finger quite literally as a "hook".  And when lip trills or shakes occur from an exaggerated right hand vibrato, the left hand is still holding the instrument!