THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

In this paper, the author obtains the following results (1)If Taylor coefflients of a function $\w(z) = \sum\limits_{n = 1}^\infty {{A_n}{z^n}} \]$ satisfy the conditions (i)$\\sum\limits_{k = 1}^\infty {k{{\left| {{A_k}} \right|}^2}} < \infty \]$ (ii)Re$\\sum\limits_{k = 1}^\infty {{A_k}} = O(1)(n \to \infty )\]$ (iii)$\{A_k} = O(\frac{1}{k})\]$, the for any h>0 the function $\\varphi (z) = \exp \{ w(z)\} = \sum\limits_{k = 0}^\infty {{D_k}} {z^k}\]$ satisfies the asymptotic equality $$\\left| {\frac{{{{\{ \varphi (z){{(1 - z)}^{ - h}}\} }_n}}}{{{d_n}(h)}} - \sum\limits_{k = 0}^n {{D_k}} } \right| = o(l)(n \to \infty )\]$$, the case $\h > \frac{1}{2}\]$ was proved by Milin (2)If $\f(z) = z + {a_2}{z^2} + \cdots \in {S^*}\]$ and $\\mathop {\lim }\limits_{r \to 1} \frac{{(1 - {r^2})}}{r}\mathop {\max }\limits_{\left| z \right| = r} \left| {f(z)} \right| = \alpha \]$, then for $\\lambda > \frac{1}{2}\]$ $$\\mathop {\lim }\limits_{n \to \infty } \frac{{\left| {\left| {{{\{ {{(\frac{{f(z)}}{z})}^\lambda }\} }_n}} \right| - \left| {{{\{ {{(\frac{{f(z)}}{z})}^\lambda }\} }_{n - 1}}} \right|} \right|}}{{{d_n}(2\lambda - 1)}} = {\alpha ^\lambda }\]$$