I'm trying to prove the following inequality: − x 1 2 + . . . + x n 2 n ≤ x 1 + . . . + x n n ≤ x 1 2 + . . . + x n 2 n 15 , ( x 1 , . . . , x n ) ∈ R The exercise seems very simple but I have problems in solving it. I was thinking about using the Cauchy-Schwarz inequality | u ⋅ v | ≤ ‖ u ‖ ‖ v ‖ but I'm not sure if it is correct. Any suggestions?

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I&#039;m trying to prove the following inequality:  −                              x          1          2                +        .        .        .        +                  x          n          2                    n        ≤                    x        1            +      .      .      .      +              x        n              n    ≤                              x          1          2                +        .        .        .        +                  x          n          2                    n                  15        ,  (      x    1    ,  .  .  .  ,      x    n    )  ∈      R  The exercise seems very simple but I have problems in solving it. I was thinking about using the Cauchy-Schwarz inequality   |  u  ⋅  v  |  ≤      ‖    u    ‖        ‖    v    ‖   but I&#039;m not sure if it is correct. Any suggestions?

elverku7 · Accepted Answer

By C-S      n          ∑              k        =        1            n              x      k      2        =            ∑              k        =        1            n              1      2              ∑              k        =        1            n              x      k      2        ≥                    (                  ∑                      k            =            1                    n                          x          k                )            2        =      |          ∑              k        =        1            n              x      k        |    ,which gives      |                            ∑                      k            =            1                    n                          x          k                    n        |    ≤                              ∑                      k            =            1                    n                          x          k          2                    n        ,which is your inequality.

cvnoticiasdg · Accepted Answer

Hint: Consider the Cauchy-Schwarz inequality in relation to the vectors  u  =  (      x    1        /    n  ,  ⋯  ,      x    n        /    n  )  ,    v  =  (  1  ,  ⋯  ,  1  )  .

Does -sqrt((x_1^2+...+x_n^2)/(n)) <= (x_1+...+x_n)/(n) <= sqrt((x_1^2+...+x_n^2)/(n)) ,(x_1,...,x_n) in RR?

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