星像的典型实照函数

<正> 称这处函数 f(z)为单位圆上的一个典型实照函数,记其全体为 T_r.设 f(z)是 T_r中的一个函数,f(z)映照|z|<的像是区域 D,如果对于 D 中任意一点都可以用完全落在 D 中的直线段与ω=0相联结的话,称 f(z)是一個星像的典型实照

TYPICALLY REAL FUNCTIONS HAVING STARLIKE IMAGES

Let the function(?)with real coefficients α_υ~((κ))be regular in |z|<1,and be such that f_κ(z)mapsthe circle |z|=γ into α contour which passess through the real axles in exactly2κ points for every γ in the interval(?).Denote thefamily of all such functions by Т_γ~κ,and write Т_γ for Т_γ~1.Any function f_κ(z)of the subclass T_γ~κ of T_γ~κ,if is such that for each γ in 1-δ<γ<1,the radius vector(?)turns continuously in the counterclockwise direction and makes κ complete revolutions as θ varies from 0 to 2π.We prove in the present note that any f_κ(z)of T_γ~κ can be represented inthe form(?)with an increasing function α(θ)satisfying(?).By means of thisformula,we establish the following:Theorem.If f_κ(z)∈Τ_γ~κ,then(?)All these estimates are precise.A function(?),with real coeffi-cients is said to be belonging to the subclass T_γ~κ(ρ,р),if there exists αρ(0<<ρ<1)such that(?)The coefficients of such a function f_(κ,р)(z)satisfy the inequality(?)This estimate is precise.The result holds good even when α_(κ+рn)~((κ))(ρ,р)are notall real,if f_(κ,р)(z)is κ-valent and(?)