I see how you’re approaching it Riccardo, but “Ebm9 with some omitted notes” is more complicated to figure than Db/Eb when you are in fact in the key of Db major. The chord’s third cannot fall between its 3rd and its 5th, or 5th and 7th; that’s a third in intervallic distance, but it’s the chord’s 3rd and 5th, or 5th and 7th, not the root and third. It’s easiest to figure out at a basic level which notes have a third interval relationship to them to get your root starting with your bass note, like you said, but then look to see if any of the other notes in the harmony exist at a third from it in the smallest interval. If not, it’s probably not the root.
So, from the notes Chris gave, my approach is, Eb in the bass. Eb to F is a major second. Eb to Db is a minor seventh. No thirds from Eb. Then F to Db is a sixth, the inversion of which is a major third. You have a Db major with an altered bass of Eb. This makes more sense because you’re in the key of Db major. Like I said, though, you need the third to give the chord its major or minor tonal function, so calling it an Ebm9 (no 3 nor 5) doesn’t work because there’s no third to indicate it’s Eb major or Eb minor. And interestingly, if you invert the Eb to the top, you still get a Db chord–Db9 (no 5 no 7).
Certainly, if you wanted to name it Ebm9, you could, but you could just as easily call it Eb9 or Dbsus2 or any other chord that contains those notes like Mike points out. It all depends on the chord around it and what you want to do with it. But, without the third relationship with Eb, you can’t tell. However, you’ve got the third with Db and F so the simpler name I think is Db/Eb.
Always interesting to see how other people process and understand music. I think that’s a big aspect on what gives a composer their own style–how they understand music and how it works and how they apply it to create the music that “sounds” good to them