What are some important mathematics results that were first developed in Economics?
The Envelope theorem is arguably one, https://en.wikipedia.org/wiki/Envelope_theorem . I would say Blackwell's papers on comparisons of experiments are a significant mathematical contribution--extending Hardy,Littlewood and Polya's earlier results, and paving the way for further investigation.

Econometricians have made statistical contributions as well; see e.g. https://en.wikipedia.org/wiki/Heckman_correction .

More recently, I saw a paper presented last year on how a certain type of stochastic PDE could be solved by reinterpreting it as a (stochastic) differential game. I would expect to see more related to this (the relationship between differential games and stochastic pdes) in the coming years, especially given the recent surge of interest in mean-field games.