UNUNIQUENESS OF SOME HOLOMORPHIC FUNCTIONS

In the present paper, we show that there exist a bounded, holomorphic function $\f(z) \ne 0\]$ in the domain $\\{ z = x + iy:\left| y \right| < \alpha \} \]$ such that $\f(z)\]$ has a Dirichlet expansion $\\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ in the halfplane $\x > {x_f}\]$ if and only if $\\frac{a}{\pi }\log r - \sum\limits_{{u_n} < r} {\frac{2}{{{u_n}}}} \]$ has a finite upperbound on $\1, + \infty )\]$, where $\\alpha \]$ is a positive constant,$\{x_f}( < + \infty )\]$ is the abscissa of convergence of $\\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ and the infinite sequence $\\{ {u_n}\} \]$ satisfies $\\mathop {\lim }\limits_{n \to + \infty } ({u_{n + 1}} - {u_n}) > 0\]$. We also point out some necessary conditions and sufficient ones Such that a bounded holomorphic function in an angular(or half-band) domain is identically zero if an infinite sequence of its derivatives and itself vanish at some point of the domain. Here some result are generalizations of those in 4].