Friday, 13 January 2017

Cauchy-Schwarz Inequality troubles





I have to prove the following inequality using the Cauchy-Schwarz inequality:
ab+c+bc+d+cd+a+da+b2
where a, b, c and d are positive real numbers.




But I am not able to do it, I am hitting dead-ends with every method I try. Please help!


Answer



By C-S and AM-GM we obtain:
cycab+c=cyca2ab+ac(a+b+c+d)2cyc(ab+ac)=2+(a+b+c+d)22cyc(ab+ac)cyc(ab+ac)=
=2+a2+c2+b2+d22ac2bdcyc(ab+ac)2+2a2c2+2b2d22ac2bdcyc(ab+ac)=2.



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