Star-center points of univalent functions

Theorem 2 provides additional information concerning the coefficient esti- mates obtained in Theorem 1. More precisely it is shown that if fx, f2 € ^ and S°f c S°f then B{f2, n) < B(fx, n), n = 1, 2, ... , where B{fx, n), B(f2, n) are the estimates for the n th coefficients of /, and f2 respectively. Finally we give examples of functions in J/^ and compare our results with those obtained in [1]. I would like to thank the referees for their helpful comments on the subject. 2. Preliminaries In thi

LEMMA 1. The set of all star center points of a function in S0 is convex. PROOF. Let g e S?, zx, z2 e D such that g{zx), g(z2) belong to Sg. We show that the segment [g(zx), g(z2)] is contained in S . Suppose [g(zx), g(z2)\ £ Sg and let w e (g(zx), g{z2)) be such that w £ Sg. Since g(zi), g(z2) are s.c.p of g(D) we have w e g(D). By the hypothesis on w there is z0 e D such that [g(z0), w] <f. g(D). Observe that if the points g(z0), g(zx), g(z2) are collinear then there is nothing to prove. Other

LEMMA 2. Let f e <9*Q, ^:Z)->5° be a univalent analytic function such that <^(0) = 0, £(£>) = S°, and let z0, zx be complex numbers such that \zo\ < |zj| = r < 1. Then the segment [/(z,), £(z0)] is contained in f(Dr), where 7)r = {z: \z\ < r}. PROOF. For £(z0) = 0 the lemma is known [2, page 220]. Let p and 6 be two real numbers such that 0 < p < I, -n < 6 < n, pe'ezx = zQ. Put <j>(z) = tf{z) + (1 - t)£(pe'ez), zeD, 0 < t < 1. Clearly <P is analytic in D, 0(0) = /(0) = 0, and for each z the poin

LEMMA 3. Let n > 2 be an integer. Given 1/2 < x < 1 and integers 1 <P <Q, define Then (1) - n\n + nFn2(x) + 2x[Fn>i(x) + 2\2Fn4(x) + ••• + (n- 2)\{n - 2)Fnn{x) + (n - 1)!(» - 1)] < 0. PROOF. We proceed by induction on n . Observe that (1) holds for n = 3. We assume that it holds for n and we prove that it holds for n + 1 . It suffices to show that the left-hand side of (1) is nonincreasing in n , for each fixed JC e [1/2, 1], or equivalently Now by the induction hypothesis we have

THEOREM 1. Let f(z) = z + Yl7=2anz" be a function in <9*0 and let a be a point of S°f. Then (i) 0 < n(f, a) < 1. (ii)//>(/, a) = 1 then \an\ < 1, n = l,2,.... (iii) n(f, a) = 1 if and only if S°f = f(D). (iv) Ifn(f, a) < 1 then \an\ < An(f, a) + R^o) = Mn(f, a), n>2, *) = l + (n-l)f\n k=2{k-l)/(k-o), a= l/(l+n(f, a)), and -n\n

THEOREM 2. Let fx, f2 be functions in S^. Let B(fx, n), B(f2, n) be the corresponding bounds on the Taylor coefficients of fx and f2 respectively, as these are defined in Theorem l(v). Suppose S*l c S^ . Then (23) B{f2,n)<B{fx,n). PROOF. Let a G 5° . Let Gx be the function obtained from /, exactly the same way as G was obtained from / in (13). Similarly, since a also belongs to 5° , let G2 be the function obtained from f2 . We have Gx (D) = S°A c S° = G2(D), Gx (0) = G2(0) = 0, and both Gx and G