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I would like to obtain a bound from below for the function
$$f(p) = \frac{pe - (1-p)^{d+1}v}{p^{v-1}}$$ subject to $0 < p \leq 1$ and $e,v,d > 0.$

The usual method is to check the boundary cases and the value at which the derivative is zero. Solving the equation involving the derivative is rather messy in this case, so I am looking at an elegant way to first bound $f(p)$ by some other function $g(p)$ and then find a bound for g that is as tight as possible.

Any hints on that? For $0 < p \leq 0.5$ one could use the fact that $(1-p)^{d+1} \geq exp(-2p(d+1))$ which still yields a function with an "ugly" derivative.

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Do you mean $0\lt p \le 1$? –  Ross Millikan Feb 6 '11 at 23:40

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