Dear friends,

I could not obtain proof for either one of the following inequalities:

or

I tried to prove it by using their tangents, but it seems not to be so simple.

Any ideas?

Thanks.

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- September 8th 2008, 07:36 AMbkarpuzExponential inequalities
Dear friends,

I could not obtain proof for either one of the following inequalities:

or

I tried to prove it by using their tangents, but it seems not to be so simple.

Any ideas?

Thanks. - September 8th 2008, 09:23 AMLaurent
There's probably a nicer proof, but here is one anyway:

Let . The derivative of the function is , which is equal to 0 at , and respectively negative and positive at the left and right of this point. As a consequence, for every , which is exactly what we want.

Laurent. - September 8th 2008, 09:45 AMbkarpuz
Thanks, Laurent.

You have any ideas for the second one?

I guess that it is a little bit harder because your function will be same for this inequality too.

To be honest, I attempted to solve the latter one at first... - September 8th 2008, 10:10 AMwingless
What do you mean? That second one is the same as the first one.. Or are you talking about another question somewhere else?

- September 8th 2008, 10:16 AMbkarpuz
- September 8th 2008, 10:21 AMwingless
Selam =)

So both inequalities are actually identical.

Also notice that the question says 'or' between the inequalities... - September 8th 2008, 10:36 AMbkarpuz
I have to say, that seems to be a joke :S

How can i miss that thing!!!

By the way, I put the 'or' there... - September 10th 2008, 10:37 PMbkarpuz
Now, I see that I always had mistakes in the calculation. (Worried)

I again checked my way for the last time and saw that it works.

Let me explain it below.

Set , we clearly know that for every since is convex.

What I need to find is that

and

Solving (1), we get

Clearly, (3) satisfies (2), and thus the solution is complete.

I really still don't know where I had mistakes all the time. (Headbang)