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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
10 months, 3 weeks 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 $$c$$? Staff · 10 months, 2 weeks ago