How to prove this ln inequality? I have the

Jasiah Vincent

Jasiah Vincent

Answered question

2022-04-08

How to prove this ln inequality?
I have the following inequality, which (supposedly) holds for every xR
1+xln(x+1+x2)1+x2
I've been struggling to find some known inequalities involving logarithms that could be applied here (and lack of condition for x to be non-negative doesn't help either). I would be very helpful for hints on how this should be approached.

Answer & Explanation

gil001q4wq

gil001q4wq

Beginner2022-04-09Added 11 answers

Just to illustrate the trick for differentiating in my comment, using the same f(x) as Norbert but writing y=1+x2
f(x)=1+xlog(x+y)y
and knowing that
dydx=xy
we compute:
f(x)=0+log(x+y)+x(1+xy)x+yxy
And simplify ... which works neatly essentially for the same reason that Davide Giraudo's substitution works, to give first and second derivatives which are easy to work with:
So we get
f(x)=log(x+y)
and
f(x)=1+xyx+y=1y
We find that there is a minimum at x=0 with
f(0)=0 and f(x) positive throughout, which gives us what we need ...

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