ON A GENERALIZATION OF CLOSE-TO-CONVEXITY OF COMPLEX ORDER

Theorem 2 .1. f E 五(b) if and only if J'(z) = (Kf (z)) 钅+! , - -- , 、、k l' where K1 and K2 are close-to-convex Junctions of complex order b.

Theorem 2.2 Let O < 柘＜転 Then 五(b1) C Tk(奶）．

Theorem 2.3. If f E 五(b) and O~r < l, l2b - 11 < 1, then for B2 > 81, 81, 02 E [O, 21r),「 2 Re{ 1 + reief"(r护 ） k 2lblr 。 i f'(r詞）}dB > -271" + 2 cos-1 1 - l2b - llr2.

Theorem 2.4. Let J E Tk(b), j2b - 11~1. Then I arg J'(z) I~k sin- l r + sin-1 _ . _. _ . - 2jbjr Next we prove a distortion theorem for Tk(b).

Theorem 2.5. Let f E Tk(b). Then (1 - l2b - ljr)(l 一r) 钅尸1 (l + r)t-1(1 + j2b - ljr) (l+r)~+2 ~IJ'(z)I~ (l-r)t+2 The equality is attained for the function fO E Tk (b) defined by (1 + 如）差1 !~ 位）= (1 + (2b - 1)01z), 101 I = 1021 = 1. (1 - 知） ~+2 The proof is immediate when we use the distortion theorems 阮g E Vi, see 13 and for

Theorem 2.6.(Covermg theorem). The image of E under functions in 九(b) contains the schlicht disc 囯 ＜k+l 一丨2b- 11 k(k + 2) .

Theorem 3.1. Let- f E Tk(b), k > 3 and be given by (1.1). Then, form= 0, l, 2, ... , there are numbers 咋m and Cmµ(µ= 0, ... , m) that satisfy lcmol = lcmml = 1, and 00 芝'Yk ::S: 3, k=O 2 0 :S;,m :S; m+l (3.1) such that f Cmµan+µ= 0(1).n戸 尸， mu=O (n---+ oo). The bounds (3.1} are best possible.

Lemma 3.1. [16] Let 術＜恥＜．．．＜恥< ()1 + 21r and let >.1, ... , >.q be real, >. > 0, 入 ：：：：入i(j = 1, ... ,q). If (3.5) q 00 心(z) = fI(l-e 一iB; z)-入，＝芷加Zn j=l n=l then as n ---+ oo. bn = 0(1)·n>.-I We now complete the proof of theorem 3.1. We write m 転

Lemma 3.2. [6] Let p E P for z EE. Then, for A> l, 「 jp(rei0)1..\dB < c(,\)~三 。 (1 - r)

Lemma 3.3. [12] Leth E P(b) in E and be given by (1.2). Then 1 沅 莊lo lh(rei平d{) :s; 1 - r2 . 1 + (4lbl2 - l)r2 Using these lemmas in (3.8), we have for k > 3 (n + m)lamnl = 0(1)·(1 - r)-TJ=-0三t-1) (r -t 1), which implies amn = O(l)n戸 尸 (n -too).

Theorem 3. 2. Let.J E 五(b), k > 3 and f be given by (1.1). Then, for q~l,n~l, Hq(n) = 0(1)·n2+(~-2)q_ The exponent [2 + (钅- 2q] is best possible. , In particular, for q = l, k > 3 H1(n) =an= 0(1)·n 钅 (n-+ oo).

Theorem 3.3. Let f E 五(b), b real, k + 2lbl·> 3. Then, for q~l,n~l Hq(n) = 0(1). n2+q(!+lbl-3). 4. Some Radii Problems In the following we find the radius of convexity for f E Tk(b).

Theorem 4.1. Let f E 五(b). Then f maps lzl < ro onto a convex domain, where r0 is the least positive root of the equation T(r) = (1 + Reµ) - (1 + k)(l + Reµ)r - (1 + k)(l - Reµ)r2 + (1 - Reµ)r3 = 0, (4.1) whereµ= 早and Reµ2: 0.

Theorem 4.2. Let f E 五(b) and F be defined, for O < a < 1, by 1 z F(z) = 云z1-i 1~i-勺(~)d~

Theorem 4.3. Let f E Tk(b) with respect to h E Yk. Let g E Vk and for a, (3 positively real with a+ (3 = l, let F(z) = J (!1 (~))°'(g'(~))f3 d~ 。 ． and z H(z) = j (h1(釧））°'(i(釧））債 。

Theorem 4.4. Let f E Vk and let F(z) = bz2-t[zt-1J(z)]'. Then FE Tk(b) for all lzl < h r1 w ere r1 zs given by (4.2). This result is sharp.

Theorem 4.5. Let F E T2(b) and let, for O < .\ < l, J(z) = (1 - .\)F(z) + .\zF1(z). Then J E T2(b) for lzl < T>., where 八= [?).+~三l This result is best possible.

Definition 1.2. Let f be analytic in E and be given by (1.1). Th細/ E Tk(b), k 2: 2, b # 0 (complex), if and only if, there exists a function g E 凶such that 阜旦E P(b) for z EE. g (z) We note that 五(1) = Tk, a class of analytic functions introduced and studited in [9] and T2(l) is the class K of close-to-convex functions. Also 花(b) = K(b) consists entirely of close-to-convex functions of complex order introduced in [1] by Al-Amiri and Fernando. 2. Some Basic Properties of 五(b)