the exponential inequality

    \[ \boxed{x^\epsilon<1+\epsilon(x-1)}\]

where x>0 and 0<\epsilon<1.

When \epsilon>1, \quad x^\epsilon>1+\epsilon(x-1)

Derivation from the AM-GM inequality:

Using a set of m positive numbers, n of which have the value a, and the m-n others have the value 1:
The arithmetic mean is \quad \displaystyle\frac{n\times a + (m-n) \times 1}{m}=\frac{n(a-1)+m}{m}, so

    \[\sqrt[m]{a^n}<1+\frac{n}{m}(a-1)\]

Or, writing \epsilon in place of \frac{n}{m},

    \[a^\epsilon<1+\epsilon(a-1)\]

exp inequal

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