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Show two inequalities - (context gamma function converges)

  1. Jan 12, 2017 #1
    1. The problem statement, all variables and given/known data

    I'm not after another proof.
    I've just got a couple of inequalities I don't know how to show when following a given proof in my book.

    These are:

    Q1) ## 0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1} ##

    So this is obvioulsy true, however I think I'm being dumb because surely this is true for all ##x##

    (the integral over the gamma function is split into ##\int ^{\infty} _1 ## and ## \int^1_0 ##) and so this inequality is used in the latter, but I don't see why it can't be used in the former, e.g ##2^{t-1}/e^{2} \leq 2^{t-1} ## is also true isn't it? and as ## x \to \infty ## this is also true, because the LHS ## \to 0 ##..

    Q2) that ##x^{t-1}e^{-x}\leq x^{-x/2}##, for ##x \geq 1##
    I am unsure how to show this, or understand why it holds,

    and then secondly I need to show that ##x^{t-1}e^{-x}\leq x^{-x/2} \iff x^{t-1}\leq e^{x/2}##

    I can write ## x^{-x/2} = e^{(-x/2) ln (x) } ##
    I am unsure what to do next or whether this helps


    2. Relevant equations

    see above

    3. The attempt at a solution

    see above
     
    Last edited: Jan 12, 2017
  2. jcsd
  3. Jan 12, 2017 #2

    pasmith

    User Avatar
    Homework Helper

    [tex]\int_1^R x^{t-1}\,dx = \begin{cases} \ln R & t = 0, \\
    \frac{R^t - 1}{t} & t \neq 0\end{cases}.[/tex] For [itex]t \geq 0[/itex] these do not converge as [itex]R \to \infty[/itex]. You're trying to prove convergence so this bound is of no assistance.
     
  4. Jan 15, 2017 #3
    thank you, makes sense !
    Q2 anyone?
     
  5. Jan 18, 2017 #4
    bump
     
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