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Math Help - help with inequality

  1. #1
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    help with inequality

    Can someone please show me how to prove this inequality:

    Let a > b > 0 and let n be a natural number greater than or equal to 2. Then
    a^(1/n) - b^(1/n) < (a-b)^(1/n).

    Thank you for your help. It is greatly appreciated.
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  2. #2
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    Quote Originally Posted by jamesHADDY View Post
    Can someone please show me how to prove this inequality:

    Let a > b > 0 and let n be a natural number greater than or equal to 2. Then
    a^(1/n) - b^(1/n) < (a-b)^(1/n).

    Thank you for your help. It is greatly appreciated.
    We can prove something stronger. Let 0<r<1 be a real number. Let a,b>0. Then -1<r-1<0 thus (a+x)^{r-1} \leq x^{r-1} for 0\leq x\leq b. This means \int_0^b (a+x)^{r-1} dx \leq \int_0^b x^{r-1} dx \implies (a+b)^r \leq a^r+b^r.

    Now if a > b > 0 \implies (a-b)>0 \mbox{ and }b>0. Using the above result it means a^r = ((a-b)+b)^r < (a-b)^r + b^r \implies a^r - b^r < (a-b)^r. Thus, if n\geq 2 then  0< 1/n < 1 and the above result holds.
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  3. #3
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    Thank you for answering. Is there a little bit easier way to show this (possibly without integration)? Maybe with sequences or mean value theorem?
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