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 $[w(z) = sumlimits_{n = 1}^infty {{A_n}{z^n}} ]$ satisfy the conditions(i)$[sumlimits_{k = 1}^infty {k{{left| {{A_k}} right|}^2}} < infty ]$(ii)Re$[sumlimits_{k = 1}^infty {{A_k}} = O(1)(n to infty )]$(iii)$[{A_k} = O(frac{1}{k})]$, the for any h>0 the function $[varphi (z) = exp { w(z)} = sumlimits_{k = 0}^infty {{D_k}} {z^k}]$ satisfies the asymptotic equality$$[left| {frac{{{{{ varphi (z){{(1 - z)}^{ - h}}} }_n}}}{{{d_n}(h)}} - sumlimits_{k = 0}^n {{D_k}} } right| = o(l)(n to infty )]$$,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}]$$$[mathop {lim }limits_{n to infty } frac{{left| {left| {{{{ {{(frac{{f(z)}}{z})}^lambda }} }_n}} right| - left| {{{{ {{(frac{{f(z)}}{z})}^lambda }} }_{n - 1}}} right|} right|}}{{{d_n}(2lambda - 1)}} = {alpha ^lambda }]$$