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MathGroup Archive 2005

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Question about "Reduce"

  • To: mathgroup at smc.vnet.net
  • Subject: [mg61888] Question about "Reduce"
  • From: "Stanley Rabinowitz" <stan.rabinowitz at comcast.net>
  • Date: Thu, 3 Nov 2005 04:59:02 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Consider the inequality:
ineq =  L(1+u^2)(1+v^2)<=u+v+M u v(u v-1)

I am trying to find all real values of L and M that make this
inequality
true for all real u and v subject to the constraints:
0 <= u <= Sqrt[3]
0 <= v <= Sqrt[3]
0 <= (1 - u v)/(u + v) <= Sqrt[3]

I thought I could do this with Reduce.

I issue the command

Reduce[ForAll[{u, v}, 0 <= u <= Sqrt[3] && 0 <= v <= Sqrt[3] && 0 <= (1
- u v)/(u + v)<= Sqrt[3], ineq],
     {L,M}, Reals] // FullSimplify

and I get back a result equivalent to

L<=0 && M <= 3Sqrt[3]-8L

Am I doing something wrong?
It is telling me that L must be less than 0.
But I know for a fact that
M = 1 + Sqrt[3] and  L = (2Sqrt[3] - 1)/8 work
and in this case, L>0.

I can verify that these value of L and M work by issuing the command

Reduce[ForAll[{u, v}, 0 <= u <= Sqrt[3] && 0 <= v <= Sqrt[3] && 0 <= (1
- u v)/(u + v)<= Sqrt[3],
     ineq/. {M -> 1 + Sqrt[3], L -> (2Sqrt[3]-1)/8}]]

which returns the value "True".

So why is it telling me L<=0?


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