Meromorphic starlike univalent functions

LEMMA 2.1. If w(z) e 8 , then for \z\ < 1 , (2.4) \z»>(z) -w(z)\<-

THEOREM 2.2. If p(z) e ?(A,B) , -(1+B)/(A-B) < y < 1 , then on \z\ = v < 1 , -[ U-2y)A-B]r+yA2r2 (1+Ar) (1+Br) — A-B (L1K1)k-(l-ABv2)\, R9 > R- where = (1+Ar)/(1+Br) , Lj = (1+A) (1-Ar ) , = y(A-B)(l-r2) + (1+B)(l-Br2) . The result is sharp.

THEOREM 2.3. 1/ p(z) e PC^BJ , y < 1 , then on \z\ = r < 1 Y v- [ Ci-2YM-g] r + yA2r2 (1-Ar)(1-Br) A-B (A-B)(l-r2) R3 ~ R4 > 3 R, ± R, 3 4 3 where = (1-Ar)/(1-Br) , L2 = (1-A) (1+Ar2) , = (1-B)(l+Br2) - y(A-B)(l-r2) . The result is sharp.

THEOREM 3.1. The radius of convexity of I (A,B) is given by the smallest root in (0,1] of (i) A2r + (A+B)r +1=0, for R£ Z RJ , (ii) (4A2+3A+B)r4 - 2[ 2(1+A)2+A-B]r2 + 4+3A+B = 0 , for R2 Z R2 , i?2 j flg being as given in Theorem 2.2 with y = 1 . https://doi.org/10.1017/S0004972700002112 Published online by Cambridge University Press

COROLLARY 3.2. The radius of convexity of l(a) is a = {[2+5o.+2a2-2(l+a)(l+a2)3s]/a(4a+3)YS . "-(a.) z e A ,

COROLLARY 3.3. Let f(z) e i*(A,B) ; then on \z\ = r < 1 , T1(H**)12-A)'B ± \f(z)\ , r-Jr;-5r/s-^;/B , if B * o , r~2exp(-Ar) < \f(z)\ ^ r 1exp(Ar) The function f(z) defined by _ zf'(z) 1+Az f(z) 1+Bz if B = 0 . s e A shows that the bounds are sharp. l'Cs-'l for two subclasses of V*(A.B) (l-2a,-l) and \ [ a] = I fa,-a,) . We next derive bounds for namely, J

THEOREM 3.4 Let f(z) e f , 6 = l-2a ; then on \z\ = r < 1 \f'(*) -2,. 2,-1 r (1-r ) a = 0 , 4r 1< r where 4/5 < an < 1 : https://doi.org/10.1017/S0004972700002112 Published online by Cambridge University Press

Theorem 2.3 with A = l-2a , B = -1 , y = 1 is equivalent to the inequality 2 + (l+2a)r + (l-2a)r3 s: 0 , which always holds for 0 < r < 1 , 0 < a < 1 . Hence there is only one case, R. <. R- , for the upper bound of Re{p(z)-zp '(z)/p(z)} with p(z) e ?a . This result applied to (3.2) gives 2 l o g \z 2f'(z)\ > - 2 2 2 2

THEOREM 3 . 5 . If f(z) e l*[a] > then on \z\ = r < 1 The results are sharp.