Let-a-b-c-gt-0-and-a-b-b-c-4-Prove-that-1-a-1-b-1-c-b-ca-27-8-Found-by-WolframAlpha-

Question Number 144676 by loveineq last updated on 27/Jun/21

Let a, b, c > 0 and (a + b) (b + c) = 4 . Prove that

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{b}{c a} ⩾ \frac{27}{8}

(Found by WolframAlpha)

Answered by ArielVyny last updated on 27/Jun/21

s u p p o s e a b c ⩽ 1

27 a b c ⩽ 32

\frac{27}{8} a b c ⩽ 4

\frac{27}{8} ⩽ \frac{4}{a b c}

o r 4 = (a + b) (b + c)

\frac{27}{8} ⩽ \frac{c b + a c + a b + b^{2}}{a b c}

\frac{27}{8} ⩽ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{b}{a c}

t h e n \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{b}{a c} ⩾ \frac{27}{8}

Commented by loveineq last updated on 28/Jun/21

n i c e a n d t h a n k s .