Max Thursday meets with a prospective client in a hotel room - not the pretty young thing he was probably hoping for, but an older woman, clutching an antique music box to her chest. She wants Thursday to deliver said box to someone else, and whenever the private detective asks for particulars, the old woman snaps at him trying to get all the details down. For good reason: Once she's outlined Thursday's directives completely, the music box drops from her fingers, and she falls dead to the floor. The old bird had been fatally stabbed shortly before Thursday arrived and she was trying her best to remain conscious and give him his orders before giving up the ghost.
That's dedication. And your hook for the book.
The scene almost repeats itself when Thursday delivers the music box to an Austrian art expert in his hotel room. At one point, the expert excuses himself to his bedroom. After a few minutes of waiting, Thursday follows him only to find the guy kiboshed - his throat slashed.
In both instances, the murderer has phoned the police in an attempt to frame Max.
A decent mystery. What's notable is that Thursday doesn't carry a gun (at least in this book). He deliberately let his license expire shortly beforehand.