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Alexandre Eremenko
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First of all, upload to the arxiv. Then you can write to the experts you know, and inform them that you uploaded your new paper. Then wait a little to give them opportunity to send you comments. If there is an possibility, give seminar/conference talks, but it is not necessary to do this before you send it to a journal.

Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429