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    Uniform convergence of $\sum|a_n|$ on compacta implies that of $\sum|a_n'|$

    P-OFKQJ

    Area
    Complex Analysis
    Topics
    Cauchy Estimates, Holomorphic Functions, Series of Functions, Uniform Convergence
    Status
    draft

    Let \(a_n(z)\) be an analytic sequence in a domain \(D\) such that \(\displaystyle \sum_{n=0}^\infty |a_n(z)|\) converges uniformly on bounded and closed sub-regions of \(D\). Show that \(\displaystyle \sum_{n=0}^\infty |a'_n(z)|\) converges uniformly on bounded and closed sub-regions of \(D\).

    Solution

    • Show a stronger statement: if \(f_n\to f\) uniformly with each \(f_n\) holomorphic on every compact subset of \(D\) then \(f_n'\to f'\) on every compact subset of \(D\).

    • We have \(\norm{f_n-f}_{\infty, D}\to 0\), the sup norm on \(D\).

    • Pick a \(\gamma\) in \(\interiorof{D}\)

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