polynomials associated with starlike univalent functions - Springer Link

Numerical Algorithms 3 (1992) 383-392

383

PC-fractions and Szeg5 polynomials associated with starlike univalent functions Frode Rcnning

Trondheim Collegeof Education, Rotvoll all~, N-7050 Charlottenlund, Norway

Each class S,~, 0~ a. This is a Carath6odory function, and it is easily verified that 1 + (1 - 2 a ) z F~(z) =

1-z

oo

= 1+ 2(l-a)

~] z".

n=l

The function F~(z) can b e regarded as a function characteristic for the class S,,. The functions of S~ are widely studied, and for information about these functions we refer to [1] and [2]. W e could mention that they were first introduced, as a generalization of starlike functions (So), by R o b e r t s o n in 1936 [7]. Carath6odory functions play an important part also in problems from signal theory, in m o m e n t problems and in constructing quadrature formulas. In this context Jones et al. [4] introduced a certain continued fraction (positive PC-frac9 J.C. Baltzer A.G. Scientific Publishing Company

F. ROnning / Starlike univalent functions

384

tion), and they proved that to every positive PC-fraction corresponds a unique Carath~odory function, and vice versa. This correspondence is such that the even approximants of the PC-fraction converge to the Carath~odory function in the open unit disk. We shall be concerned with positive PC-fractions corresponding to Carath6odory functions with real Maclaurin coefficients and normalized by F(O) = 1. Then the PC-fraction can be written 2 __

1

(1 - a 2 ) z

1

(1 -822)z

61

-1- 62Z -1-

62

- -

1 + ~1Z -b

-1O,

r~(0,1>.

Now, R , is the unique solution in (0, 1) of Gn(r)= 0, and from the above it follows that Rn+ 1 > R n. H e n c e , {Rn} is monotonically increasing, and since it is b o u n d e d by 1, the limit is 1. Let -r < 1 be given. W e shall prove that, for n sufficiently large, r n > ~-. This will prove that also lim n _~=r,, = 1. Choose n o to be the smallest value of n with R n > r. If r,o > -r, we are done, so assume rno < ~" < Rno. Pick a point z, [ z [ = ~-, on /'no. T h e n

{ z - ( 1 + 18,,o{)1 = 18"~ ,./-no

Using the triangle inequality we get

{8.o{

- q-no -

~