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Question Number 140959 by loveineq last updated on 14/May/21

Let a,b ≥ 0. Prove that (1/4)∙(((2+a)(2+b))/((1+a)(1+b))) ≥ ((4−a−b)/(4+a+b))

Leta,b⩾0.Provethat14⋅(2+a)(2+b)(1+a)(1+b)⩾4−a−b4+a+b

Answered by MJS_new last updated on 14/May/21

this is easy a, b ≥0 ⇒ 1+a>0∧1+b>0∧4+a+b>0 ⇒ (2+a)(2+b)(4+a+b)≥4(4−a−b)(1+a)(1+b) ⇒ 5a^2 b+6a^2 +5ab^2 +6b^2 ≥0 which is true since a, b ≥0

thisiseasya,b⩾0⇒1+a>0∧1+b>0∧4+a+b>0⇒(2+a)(2+b)(4+a+b)⩾4(4−a−b)(1+a)(1+b)⇒5a2b+6a2+5ab2+6b2⩾0whichistruesincea,b⩾0

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