Regarding the general question, it seems to me a slightly more pressing question is whether the result is well-known or not ("known" is trickier; various things were known at some point and forgotten to various extents, and it may not be a bad idea to republish such things), or otherwise easy enough to deduce using known techniques. The attitude I think is appropriate here is one of humility. Just consider the fact that smart people have been doing mathematics for thousands of years, and in this particular case smart people have been doing complex analysis for centuries. For relatively old fields all of the easy results are likely to have been proven already, or at least that that should be the default assumption. To assume anything else seems to me a little arrogant.
For example, a few years ago I went through the following several times in a row:
- Observe some curious combinatorial statement that did not seem to be well-known.
- Later discover that it is somewhere in Richard Stanley's Enumerative Combinatorics.
So there were two options: either the trains of thought I had been pursuing had already been well studied, or Richard Stanley is a mind-reading time traveler.
Anyway, the only advice I can give about what to do in this situation is to become quite familiar with the basic results in the field. Then maybe talk to a trusted mathematician and ask whether the result sounds familiar or not. Perhaps pose the question you answer (without your answer) on math.SE and see if it's easy enough for someone to answer in their spare time.