How to show ( a + b ) n ≤ a n + b n , where a , b ≥ 0 and n ∈ ( 0 , 1 ]?Does anyone happen to know a nice way to show that ( a + b ) n ≤ a n + b n , where a , b ≥ 0 and n ∈ ( 0 , 1 ]? I figured integrating might help, but I've been unable to pull my argument full circle. Any suggestions are appreciated :)

Question

How to show   (  a  +  b      )    n    ≤      a    n    +      b    n  , where   a  ,  b  ≥  0 and   n  ∈  (  0  ,  1  ]?Does anyone happen to know a nice way to show that   (  a  +  b      )    n    ≤      a    n    +      b    n  , where   a  ,  b  ≥  0 and   n  ∈  (  0  ,  1  ]? I figured integrating might help, but I&#039;ve been unable to pull my argument full circle. Any suggestions are appreciated :)

slapadabassyc · Accepted Answer

We have  1  =      a          a      +      b        +      b          a      +      b        ≤            (              a                  a          +          b                    )        n    +            (              b                  a          +          b                    )        n  for   0  &amp;lt;  n  ≤  1.

Avery Stewart · Accepted Answer

The cases   a  =  0 is trivial. For   a  ≠  0 let   m  =  1      /    n and   c  =      a          1              /            m        =      a    n   and   d  =      b          1              /            m        =      b    n    . Then  (  a  +  b      )    n    ≤      a    n    +      b    n      ⟺    (      c    m    +      d    m        )          1              /            m        ≤  c  +  d    ⟺      ⟺        c    m    +      d    m    ≤  (  c  +  d      )    m      ⟺    1  +  (  d      /    c      )    m    ≤  (  1  +  d      /    c      )    m      ⟺      ⟺    (  d      /    c      )    m    ≤  (  1  +  d      /    c      )    m    −  1    ⟺      ⟺    m      ∫    0          d              /            c            x          m      −      1        d  x  ≤  m      ∫    0          d              /            c        (  1  +  x      )          m      −      1        d  xwhich holds as   d      /    c  ≥  0 and   m  −  1  ≥  0.

Does anyone happen to know a nice way to show that (a+b)^n<=a^n+b^n, where a,b>=0 and n in (0,1]? I figured integrating might help, but I've been unable to pull my argument full circle. Any suggestions are appreciated :)

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