under one’s arm—held between the arm and the body:
- Alfredo wandered about the snow-covered streets with the manuscript of his last tragedy under his arm.
under one’s hand—(of a document) authenticated by a person’s signature:
- Lady Bellasis gained so much on the duke, that he gave her a promise under his hand to marry her.
Note: The expression does not correlate in meaning with the compound underhand—(also: underhanded)
1. done in a secret or stealthy manner:
- Their influence must have been exercised in a purely underhand way.
2. not legal; corrupt or marked by deception:
- We have found that banks are employing increasingly underhand methods to avoid their responsibility.