How can we prove that the following inequality in four real variables? $$3dc^2+3bc^2+3d^2c-6dc+3b^2c-4abc-6bc+3ad^2-4abd+3a^2d-6ad+ab^2+a^2b-2ab\ge 0~,$$ with $a,b,c,d\ge 2$.

  • $\begingroup$ Could you please suggest me a helpful software tool to verify that this inequality is true? WolframAlpha is not helpful in this case. $\endgroup$ – Penelope Benenati Jul 19 '20 at 17:02
  • $\begingroup$ Why do you believe it to be true? $\endgroup$ – LSpice Jul 19 '20 at 17:10
  • $\begingroup$ Yes, I run several computation experiments and there are some other reason to believe that is true, related to the original problem where this inequality has been originated. $\endgroup$ – Penelope Benenati Jul 19 '20 at 17:11
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    $\begingroup$ i think this can be proven using am-gm and the fact that all the variables are greater than 2 $\endgroup$ – vidyarthi Jul 19 '20 at 17:14
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    $\begingroup$ Do you know when does equality occur? Did you try the substitutions $a=2+x$ etc? $\endgroup$ – Fedor Petrov Jul 19 '20 at 17:34

The inequality is false for $a=1799, b=105, c=1024, d=4$.

This counterexample was found and verified with Mathematica, as follows:

enter image description here

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    $\begingroup$ Thank you Iosif. How did you find these values? $\endgroup$ – Penelope Benenati Jul 19 '20 at 18:14
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    $\begingroup$ @PenelopeBenenati : I have added the way the counterexample was found. $\endgroup$ – Iosif Pinelis Jul 19 '20 at 18:49
  • $\begingroup$ thank you very much. $\endgroup$ – Penelope Benenati Jul 19 '20 at 19:10

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