The signature, although hurried, looks authentic — especially the final flourish. Perhaps someone who’s studied Billie Holiday’s performances, tours, and isolated club dates can tell us when she was in Marion, Ohio?
This goes back to the idyllic days when hotels provided stationery for their guests — it was, as you notice A NEW HOTEL WITH A COMPLETE SERVICE — and people used fountain pens.
I imagine that some fan thrust a blank envelope under Billie’s nose and said, “Could I have your autograph, Miss Holiday?” and she signed it standing up . . . at least the calligraphy suggests this. I’d entertain alternate scenarios from any Holiday-fanciers. eBay, of course!