# Inequality in four real variables

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 $$a,b,c,d\ge 2$$.

• Could you please suggest me a helpful software tool to verify that this inequality is true? WolframAlpha is not helpful in this case. – Penelope Benenati Jul 19 '20 at 17:02
• Why do you believe it to be true? – LSpice Jul 19 '20 at 17:10
• 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. – Penelope Benenati Jul 19 '20 at 17:11
• i think this can be proven using am-gm and the fact that all the variables are greater than 2 – vidyarthi Jul 19 '20 at 17:14
• Do you know when does equality occur? Did you try the substitutions $a=2+x$ etc? – Fedor Petrov Jul 19 '20 at 17:34

## 1 Answer

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

This counterexample was found and verified with Mathematica, as follows: • Thank you Iosif. How did you find these values? – Penelope Benenati Jul 19 '20 at 18:14
• @PenelopeBenenati : I have added the way the counterexample was found. – Iosif Pinelis Jul 19 '20 at 18:49
• thank you very much. – Penelope Benenati Jul 19 '20 at 19:10