Ununiqueness of some holomorphic functions (Q1079684)

Applying the well known Malliavin theorem the author mainly establishes the conditions under which a bounded holomorphic function admits a Dirichlet expansion, namely: Result: Suppose that \(0<u_ 1<u_ 2<u_ 3<..\). \(\lim_{n\to \infty}(u_{n+1}-u_ n)>0\). Then there exists a function f(z)\(\not\equiv 0\) which is holomorphic and bounded in \(\{z=x+iy:\) \(| y| <\alpha \}\) \((\alpha >0)\) and which has a Dirichlet expansion \(f(z)=\sum^{\infty}_{1}d_ ne^{-u_ nz}\) \((x>x_ j)\) if and only if \[ \overline{\lim}(\frac{2\alpha}{\pi}\log r-2\sum_{u_ n<r}\frac{1}{u_ n})<\infty \] where \(x_ j<\infty\) is the abscissa of convergence.