Prove or disprove that | a 1 | + | a 2 | + … + | a n | ≤ n a 1 2 + … + a n 2 by showing that R H S − L H S ≥ 0if possible.

Question

Prove or disprove that       |        a    1        |    +      |        a    2        |    +  …  +      |        a    n        |    ≤  n            a      1      2        +    …    +          a      n      2       by showing that    R  H  S  −  L  H  S  ≥  0if possible.

Josie Stephenson · Accepted Answer

Notice that       |        a    i              |        2    =      a    i    2  . Using the Generalized Mean Inequality, we see                    |                    a        1                    |            +      ⋯      +              |                    a        n                    |              n    ≤                              |                          a          1                                      |                    2                +        ⋯        +                  |                          a          n                                      |                    2                    n      which we can rewrite to      |        a    1        |    +  ⋯  +      |        a    n        |    ≤      n              a      1      2        +    ⋯    +          a      n      2      and since       n    ≤  n we get      |        a    1        |    +  ⋯  +      |        a    n        |    ≤  n            a      1      2        +    ⋯    +          a      n      2      Alternatively, we can use Cauchy-Schwarz:  (      |        a    1        |    ⋅      |    1      |    +  ⋯      |        a    n        |    ⋅      |    1      |        )    2    ≤  (      |        a    1              |        2    +  ⋯      |        a    n              |        2    )  (      |    1            |        2    +  ⋯  +      |    1            |        2    )which we can also rewrite to      |        a    1        |    +  ⋯  +      |        a    n        |    ≤      n              a      1      2        +    ⋯    +          a      n      2      and finishing the proof the same way.

George Bray · Accepted Answer

I think you mean   R  H  S  =      n    (          a      1      2        +          a      2      2        +    .    .    .    +          a      n      2        )  If so we have:  R  H  S  −  L  H  S  =                    ∑                  1          ≤          i          &amp;lt;          j          ≤          n                                      (                      |                                a            i                                |                    −                      |                                a            j                                |                    )                2                    R      H      S      +      L      H      S        ≥  0, which you wished.

Prove or disprove that <mrow class="MJX-TeXAtom-ORD"> | </mrow> a 1 </msub>

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