# Inequality of complex numbers (Solved)

 Importance: Medium ✭✭
 Author(s):
 Subject: Analysis
 Keywords:
 Recomm. for undergrads: yes
 Posted by: feanor on: April 7th, 2010
 Solved by: fedorpetrov here in comments
Conjecture   There exists a real positive , such that for any and any where for and , the following holds:

## Bibliography

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### Yes, such exists, say

Yes, such exists, say works. Assume the contrary and consider the counterexample. Without loss of generality, , else multiple all 's to some so that this bacomes true, LHS is multiplied by , while RHS only by . So, we again get a counterexample. Denote , , . Since LHS does not exceed 2, we have (else RHS is too large). Hence for s.t. . Then we have , where , by some easy estimate. Finally, LHS equals and we just use estimate for small enough ( is small enough, since ).

### A hint: \par Let with

A hint: \par Let with small and . Then and . \par Now from it follows that This shows that your inequality has the right order of magnitude'' with .

### A hint

I am the author of this hint and somehow mismanaged the posting. So here it is again:

Let with small and . Then and .

Now from it follows that This shows that your inequality has the "right order of magnitude" with .

### Requesting background information

To feanor, the author of this conjecture:

What is the motivation for this conjecture ? The selected importance "medium" let me assume the verification or falsification of this conjecture would bring some benefit. If possible, describe that benefit ("practical" applications or consequences), please.

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