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Proving all derivatives of a function are bounded by another function

  1. Jan 28, 2012 #1
    I just ran into this problem and have no idea how to solve it. Basically I'm trying to prove that all orders of derivative of the given function is bounded by the function on the right. I'm pretty sure the inequality is true, but I really have no clue on how to prove it. I thought about using Leibniz Rule (I actually didn't know about it before), but I got stuck. I think I might be making it more complicated than it actually is.

    1. The problem statement, all variables and given/known data

    Prove

    [itex]|d^n_x(\frac{e^{-(a+b)x}}{3+\cos(e^{-bx})})|≤\frac{C}{(1+x)^d}[/itex]

    for any [itex]n[/itex], any [itex]d>0[/itex], and all [itex]0≤x<∞[/itex],

    where [itex]a[/itex] and [itex]b[/itex] are constants greater than zero and [itex]C[/itex] is a constant greater than zero that depends on [itex]d[/itex] and [itex]n[/itex].
    2. Relevant equations

    General Leibniz Rule (relevant?)

    [itex]f[/itex] and [itex]g[/itex] are [itex]n[/itex] differentiable functions then the [itex]n[/itex]th derivative is of their product is

    [itex](f\cdot g)^n=\sum_{k=0}^n \begin{pmatrix}n\\k\end{pmatrix}f^{(k)} g^{(n-k)}[/itex],

    where [itex]\begin{pmatrix}n\\k\end{pmatrix}[/itex] is the binomial coefficient.

    3. The attempt at a solution

    I don't know if any of this is correct....

    The function is analytical for all x and therefore smooth for all x. Hence, for n=0, the left side exponentially decays to zero. From here, for n=0, I don't know how to prove that the inequality is true for all d>0.

    For n>0, since the function is smooth, I can apply Leibniz Rule as follows:

    [itex]f=e^{-(a+b)x}[/itex] and [itex]g=(3+\cos(e^{-bx}))^{-1}[/itex].

    Then,

    [itex] f^{(k)}=(-(a+b))^k e^{-(a+b)x} [/itex].

    Therefore,

    [itex]|d^n_x(\frac{e^{-(a+b)x}}{3+\cos(e^{-bx})})|=\sum_{k=0}^n \begin{pmatrix}n\\k\end{pmatrix}(-(a+b))^k (e^{-(a+b)x})g^{(n-k)}[/itex]

    Lastly, I guess I can argue, since the left side of the inequality is smooth and always multiplied by an exponentially decaying function, the left side is bounded.

    I'm pretty sure my last step is wrong because I didn't show what the behaviour of the g^(n-k) is. Also, again I don't know what to do for the "any d>0" part. I'm really stuck and I have no idea what to do. I'm a mechanical engineer so my math skills aren't super advanced. Thanks in advance for any help.
     
    Last edited: Jan 28, 2012
  2. jcsd
  3. Jan 28, 2012 #2
    sorry for the double post, but I was looking more into this and I think this problem relates to Schwartz space or "function space of functions all of whose derivatives are rapidly decreasing."

    "To put common language to this definition, we could note that a rapidly decreasing function is essentially a function f(x) such that f(x), f'(x), f''(x), ... all exist everywhere on the real line and go to zero as x goes to +/- infinity faster than any inverse power of x....ANY smooth function f with compact support is in Schwartz space." (according to Wikipedia)

    I'm not really sure how to use any of this to prove/disprove the inequality above. Hopefully, someone else can understand this better than me. thanks.
     
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