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Prove-that-a-b-b-c-c-a-8-1-3-ab-bc-ca-3-for-a-b-c-gt-0-

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Question Number 120215 by benjo_mathlover last updated on 30/Oct/20
Prove that (((((a+b)(b+c)(c+a))/8) ))^(1/3) ≥ (√((ab+bc+ca)/3)) for a,b,c > 0
Provethat(a+b)(b+c)(c+a)83⩾ab+bc+ca3fora,b,c>0
Answered by TANMAY PANACEA last updated on 30/Oct/20
((a+b)/2)≥(√(ab)) ((b+c)/2)≥(√(bc)) ((c+a)/2)≥(√(ac)) multiply them (((a+b)(b+c)(c+a))/8)≥(√(a^2 b^2 c^ )) (((a+b)(b+c)(c+a))/8)≥abc [(((a+b)(b+c)(c+a))/8)]^(1/3) ≥(abc)^(1/3) now(√((ab+bc+ac)/3)) ≥(√((ab×bc×ac)^(1/3) )) (√((ab+bc+ac)/3)) ≥(a^2 b^2 c^2 )^(1/6) (√((ab+bc+ac)/3)) ≥(abc)^(1/3)
a+b2⩾abb+c2⩾bcc+a2⩾acmultiplythem(a+b)(b+c)(c+a)8⩾a2b2c(a+b)(b+c)(c+a)8⩾abc[(a+b)(b+c)(c+a)8]13⩾(abc)13nowab+bc+ac3⩾(ab×bc×ac)13ab+bc+ac3⩾(a2b2c2)16ab+bc+ac3⩾(abc)13

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