Exercise 4, § 2

Functions of a Real Variable, III, 125

For every real $x$ and every number $p > 1$ prove the inequality

|1 + x|^p \leq 1 + px + \frac{p(p-1)}{2} x^2 + \cdots + \frac{p(p-1)\ldots(p-m+2)}{(m-1)!} x^{m-1} \\ + \frac{p(p-1)\ldots(p-m+1)}{m!} |x|^m + h_p\ |x|^p

where one puts

m = [p]

(the integer part of

p

) and

h_p = \frac{p(p-1)\ldots(p-m+1)}{(m-1)!} \int_0^1 z^{p-m}(1-z)^{m-1}\ dz.

Solution

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