单位圆上的有界单叶函数

<正> 1.引言设函数在单位圆|z|<1上是正则的,单叶的.它映照|z|<于|w|<1中.这种f_k(z)的全体形成一函数族 B_k,乃是 k 称的有界单叶函数族.对于 B_1中的函数 f_1(z),劳宝生讨论了|a|,|z_0|<1,|f_1(z_0)|和|f′(z_0)|四者之间的关系.利用关系式(?),他的许多结果可以直接推广到函数族B_k中来.但是关于f_k(z),还有些应该直接研讨的问题.例如当|a|,|z|取定值或|a|,

BOUNDED SCHLICHT FUNCTIONS IN THE UNIT CIRCLE

Let the functionbe regular and schlicht in the unit circle |z|<1 and in which let it besuch that |f_k(z)|<1.The totality of all such function forms a class whichshall be denoted by B_k.For a function f_1(z)of B-1 and a point z_0 of |z|<1,R.M.Robinsonhas discussed the relations between the four quantities |α|,|z_0|,|f_1(z_0)|and |f′_1(z_0)|.By means of the relation(?),some of Robinson'sresults can be extended to the class B_k.However,there are problems in theclass B_k(k>1)which are not allowable to solve them in this manner.Employ the method of parameter representation we obtain the following Theorem.Let f_k(z)∈B_k and write(?),|α|=|f′_k(0)|,r=|z|<1,then,corresponding to the three cases:1)(?)λ being the least positive root of the equation(?)2)(?)3)(?)we have respectively1)(?)with(?)2)(?)with(?)3)(?)with(?)The estimates 1). 2) and 3) are precise.