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Show there exists constant such that inequality is satisfied for all reals

I don't how to tackle with following problem:

Show there exists constant \(0<c<1 \) (depending on \(n\)) such that \( \sum_{i=1}^n x_i^3x_{i+1} \le c \sum_{i=1}^n x_i^4 \) is satisfied for arbirary reals where \( \sum_{i=1}^n x_i=0 \) and \( x_{n+1}=x_1 \).

I'll be grateful for help

Note by Thomas Johnson
2 years, 1 month ago

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What have you tried?

Are you familiar with Classical Inequalities? If yes, which do you think would be applicable?

For \( n = 2 \), what do you think is the best value of \(c\)?
For \( n = 3 \), what do you think is the best value of \(c\)?

Calvin Lin Staff - 2 years ago

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