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Stephan Ruscheweyh's Convolutions in Geometric Function Theory (Séminaire de PDF

By Stephan Ruscheweyh

ISBN-10: 2760606007

ISBN-13: 9782760606005

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S - 1). Then l+xz l+yz g E K(a,S) . 12 applies. 5. ~,v Let a ~ y a ~ ~ ~ ~ ~u~h th~ a ~ 0 ~ a-I y, a ~ v ~ S - 0 • a-I. a - E R be E T(a,S)*, g E K(y,o), F E K(~,v). 13 are in Sheil-Small [74J and in [58]. PROOF: First assume ~ ~ v. 5. Now let > R E K(~ There are functions and ~ 1 and without loss of generality assume - v,O), S E K(v,v) =R f • S. Let ~ 1. m = [~] Q = R11m such that Q E KCC~ - v)/m,O) For with v k = O,l, ... ,m H(~-v)/m c H . • m - 1. * (gQk) Multiplication of all these functions yields f f (1.

16 ) ,... 18) f E RO members map 2 - 2a f N. 17) Thus we have , R~ = S~ and R1 <-~> RO >Re < zf' 1 ~ > 2, z EU , zfll Re(--y;- + 1) > 0, z E U • = KO' KO where is the subclass of S whose U onto convex domains. The following theorem is basic for the theory of prestarlike functions. THEOREM 2. 1: PROOF: i) Let a 1 and ~ f,g E R. a Then f * g E Ra . 11). 3 - 2a)* c K(1,3 - 26)* = T(1,3 - 26)* . 19) f,g E AI' Re f z> 1 2. 21) are much stronger. conjecture of Palya and Schoenberg [42J is valid. 1, ii) implies that KO c S~ c R1 ' an old result due to Strohhacker [81J.

1). h a. * a. +l)/2a. ). (f/z) We give a slightly different proof. -1) /2a. /2' But = h a. ) * , g E = Po. /2 . ::: I (B(l,O) h a. 19 is well known (Pommerenke [43J). 4. for f = gH, g E RO' H E where and the proof is complE,te. ,O), a. > 1. 20: f~z(1-z) PROOF: obtain and f E = (£/z)a [iJ Z a. ). a F ~ (1 + z)/(1 - z)1+2a. and thus We have B(a,O) (h / z) * 1+z ::: a. (1-z) 1+2a. 4 we 67 Raising this to the n-th power and letting a:: lin gives the result. 20 has first been proved by Zamorski [94J.

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Convolutions in Geometric Function Theory (Séminaire de mathématiques supérieures) by Stephan Ruscheweyh


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