
Inequality of complex numbers (Solved)
Conjecture There exists a real positive
, such that for any
and any
where
for
and
, the following holds:







Bibliography
* indicates original appearance(s) of problem.
A hint: \par Let with
On July 29th, 2010 Anonymous says:
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
On August 1st, 2010 Christian Blatter says:
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
On April 21st, 2010 Carolus says:
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.
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
).