Prove if f is a concave function, then f ( a x 1 + b x 2 + c x 3 a + b + c ) ≥ a f ( x 1 ) + b f ( x 2 ) + c f ( x 3 ) a + b + c .

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Prove if   f is a concave function, then   f      (                  a                  x          1                +        b                  x          2                +        c                  x          3                            a        +        b        +        c              )    ≥            a      f      (              x        1            )      +      b      f      (              x        2            )      +      c      f      (              x        3            )              a      +      b      +      c        .

Tori Knight · Accepted Answer

If   f is a concave function, Jensen&#039;s inequality says that  f      (                            ∑          i                          a          i                          x          i                                      ∑          i                          a          i                      )    ≥                    ∑        i                    a        i            f      (              x        i            )                      ∑        i                    a        i            Put       a    1    =  a,       a    2    =  b,       a    3    =  c where the sum runs over   1  ≤  i  ≤  3.Note: If   f is a convex function, Jensen&#039;s inequality reverses and now says that  f      (                            ∑          i                          a          i                          x          i                                      ∑          i                          a          i                      )    ≤                    ∑        i                    a        i            f      (              x        i            )                      ∑        i                    a        i

Prove: If f is a concave function, then f (ax_1+bx_2+cx_3/a+b+c)>= (af(x_1)+bf(x_2)+cf(x_3))/(a+b+c)

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