Is there an elementary proof of this seemingly simple inequality?

https://math.stackexchange.com/questions/4990665/is-there-a-complex-variable-proof-of-ez-1-ge-1-1-e-on-the-unit-circle-z

by vector_spaces • 2 years ago

2 3 2 years ago

math.stackexchange.com

vector_spaces2• 2 years ago

This inequality has invited lots of incorrect replies, and the correct proofs are surprisingly complicated, involving difficult calculations and estimates. The asker seems to be looking for something that perhaps a (very) clever undergraduate could come up with. Curious to see if anyone here might have any luck

yzydserd• 2 years ago

The reference in the Q to e^i0 and the unit circle makes me feel like a fruitful avenue may be to pursue the fact that e^ix is cosx + i sinx and avoid anything to do with critical points. Doubly so when you recall e^i(pi) + 1 = 0 has similarity to the Q expression. e ... circles ... it has to be this path? But a) being a middle aged math newbie (thanks mathacademy.com!) I'm not yet au fait with proofs, and b) no reference to this on the stackexchange page, I am likely off track, but curious what the better avenue is if it isn't that. It almost feels like the Q is asking for a proof of the euler formula. But I am rather naive at this level of math, for now :-)

vector_spaces2• 2 years ago

I don't think those are bad instincts -- especially given that simplicity is desired! So far none of the answers on that page meet the brief, so I wouldn't read much signal from the fact that no one has approached it that way

Is there an elementary proof of this seemingly simple inequality?

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