Frank Sinatra considered himself a jazz singer, and Sinatra's behind-the-beat, ahead-of-the-beat virtuosic phrasing - the "swing" in his delivery - comes directly from Louis Armstrong who invented the style.
I can't think of a musician more relevant to a discussion of Sinatra than Armstrong.
If your talking musical, stylistic nuance fine, however I thought the discussion was Sinatra's influence as a "popular artist". In terms of "pop" star or "pop" music, Crosby was a hell of a bigger influence on all pop musicians who came after him...certainly Bing was Sinatra's biggest influence.
Crosby was truly the first "pop" singing sensation but Sinatra raised the bar.