Ratio Inequality How can I prove that, a <mrow class="MJX-Te

protestommb 2022-06-25 Answered
Ratio Inequality
How can I prove that,
a 1 + a 2 + + a n b 1 + b 2 + + b n max i { a i b i }
where 1 i n, and a i a j and b i b j , i j
Edit I have figured out that the above assumptions about a i , and b i are not needed.
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Answers (2)

kuncwadi17
Answered 2022-06-26 Author has 16 answers
The a i can be arbitrary real numbers, but the b i need to be positive. Then
(*) a j b j { max i a i b i } for  j = 1 , , n
and adding these gives the desired inequality.
If the b i are not required to be positive then the inequality must not hold, a counter-example is
2 1 3 2 > max { 2 3 , 1 2 } .
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Averi Mitchell
Answered 2022-06-27 Author has 8 answers
Suppose this holds for n = 2 (prove this base case yourself). Then
( a 1 + . . . a k ) + a k + 1 ( b 1 + . . . + b k ) + b k + 1 max ( a 1 + . . . + a k b 1 + . . . + b k , a k + 1 b k + 1 )
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