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Math Help - cauchy inequalities [complex analysis]

  1. #1
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    cauchy inequalities [complex analysis]

    hello all,..

    i need help with this problem

    If f is a holomorphic function on the strip -1 < y < 1, x \in \mathbb{R} with |f(z)| \leq A (1 + |z|)^{\lambda}, where \lambda is a fixed real number, for all z in that strip.

    show that for each integer n \geq 0 there exist A_n \geq 0 s.t

    |f^{(n)}(x)| \geq A_n(1 + |x|)^{\lambda} for all x \in \mathbb{R}.

    my uncomplete solution :

    fixed n. for any x_0 \in \mathbb{R}, we can make a circle Ccentered at x_0 with radius R < 1.

    by cauchy inequalities, we have

    |f^{(n)}(x_0)| \leq \frac{n!||f||_C}{R^n}

    with ||f||_C = \sup_{z \in C}|f(z)|.

    let \sup_{z \in C}|f(z)| = |f(z_0)|

    because |f(z_0)| \leq A (1 + |z_0|)^{\lambda}, then

    |f^{(n)}(x_0)| \leq \frac{n!A (1 + |z_0|)^{\lambda}}{R^n}

    what should I do next?

    thx for any comment
    Last edited by dedust; February 15th 2010 at 03:27 PM.
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  2. #2
    Senior Member
    Joined
    Nov 2009
    Posts
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    Quote Originally Posted by dedust View Post
    hello all,..

    i need help with this problem

    If f is a holomorphic function on the strip -1 < y < 1, x \in \mathbb{R} with |f(z)| \leq A (1 + |z|)^{\lambda}, where \lambda is a fixed real number, for all z in that strip.

    show that for each integer n \geq 0 there exist A_n \geq 0 s.t

    |f^{(n)}(x)| \geq A_n(1 + |x|)^{\lambda} for all x \in \mathbb{R}.

    my uncomplete solution :

    fixed n. for any x_0 \in \mathbb{R}, we can make a circle Ccentered at x_0 with radius R < 1.

    by cauchy inequalities, we have

    |f^{(n)}(x_0)| \leq \frac{n!||f||_C}{R^n}

    with ||f||_C = \sup_{z \in C}|f(z)|.

    let \sup_{z \in C}|f(z)| = |f(z_0)|

    because |f(z_0)| \leq A (1 + |z_0|)^{\lambda}, then

    |f^{(n)}(x_0)| \leq \frac{n!A (1 + |z_0|)^{\lambda}}{R^n}

    what should I do next?

    thx for any comment

    Cauchy inequalities

    If f is holomorphic in an open set that contains the closure of a disc D centered at z_0 and of radius R, then

    |f(z_0)| \leq \frac{n!||f||_C}{R^n},

    where ||f||_C = \sup_{z \in C} |f(z)| denotes the supremum of |f| on the boundary circle C
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