I have to prove the following inequality using the Cauchy-Schwarz inequality:
ab+c+bc+d+cd+a+da+b≥2
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)2∑cyc(ab+ac)=2+(a+b+c+d)2−2∑cyc(ab+ac)∑cyc(ab+ac)=
=2+a2+c2+b2+d2−2ac−2bd∑cyc(ab+ac)≥2+2√a2c2+2√b2d2−2ac−2bd∑cyc(ab+ac)=2.
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