On close-to-convex functions of order

THEOREM 1.1. Let β ^ 0 and m be a positive integer. Then fe K(β) iff there exists an m-fold symmetric function g e K(β/m) such that f'(z m) = g'(z) n. 263

THEOREM 2.1. Let fe K(β). Then ((1 - r)/(l + r)Y +2M{r, /') is a decreasing function of r, and hence o) = lim^ (1 — r) β+2M(r, /') exists and is finite. Ifω>0 and f is given by (1.3), then there exists θ0 such that φ\z) = (1 - ze~ iθή~ 2 and ω = lim^ (1 - ry +2 \f{re iθή \.

LEMMA 2.1 Let fe K{β) and ω = linw (1 - r)^ 21 f{re iθή \ > 0. Then given δ > 0, we may choose C = C(δ) > 0 and r0 = rQ(δ) < 1 such that for rQ ^ r < 1 we have

Theorem 2.1 and (1.3) we find, with z = re iθ, I/'(*)! = \p(z)\'\l-zr. Hence, with C > 0 and E as above, we find (1 — r)' 5 Jc(i-r) C (1 — r) β+1 and the lemma now follows upon choosing C sufficiently large. L E M M A 2.2. Let fe K(β), ω = l i m ^ (1 - r)? +2 \ f'(re ίθ«) \>Q,rn =

THEOREM 2.2. Let feK(β) be given by (1.1), and let ω = lim^i (1 — r) β+2M(r, / ' ) . Let Γ denote the gamma function. Then ' am\ co nβ Γ(β + 2) Also, if ω = l i m ^ (1 - ry +2 \ f'(re

THEOREM 2.3. Let feK(β) be given by (1.1) and let Fβ be as above. ( i ) There exists an integer n0 depending on f such that | an | g An{β) for n ^ n0. (ii) If n^β + 2, then \an\^ An{β). (iii) If β is an integer, then \ an | tί An(β) for all n. Note that since VkaK(β) with β — k/2 — 1, we have from (ii)

THEOREM 2.4. Let feK(β) be given by (1.1). If ω > 0, then \\an+1\ -\an\\ _ βω n*~ ι Γ(β + 2) The theorem is in general false when ω = 0.

Theorem 2], and thus we merely sketch the proof. We define ωn as in Lemma 2.2, λM = arg ωn, and Since ωn = [(1 - rn)p(rne iθή] β, lim^*, λΛ exists by [6, Theorem 2] As in [11, Lemma 3] we find that as n—> oo, (2.11) an — e" iθ^an^ and hence as %->oo (2.12)

LEMMA 3.1. Let fe K(β), ω = lim^ (1 - r)^ +21 f{τe idή \ > 0. Let C> 0 and λ > 0 be fixed, and forO<R<l define E = {θ: C(l - R) ^ I θ - θ01 ^ TΓ}, E' = [-7Γ, π]\E. Define ω(R) = (1 - Ry+ 21 f'(Re iθή | and as \ \f'R{Re i>)\ ιdθ~\ \f\Re JE' JE'

LEMMA 3,2. Let fe K(β), ω > 0, and fR be as above. If X(β + 2) > 1, then as r —> 1,

THEOREM 3.1. Lei fe K(β) and x(β + 2) > 1. Then lim (1 -

Theorem 3.2] we have 11 2π Hence \ 2π \f'(z)\ λdθ = Jo and since 7 < 2 we have as r —* 1 (1 _ r ) ^ ^ - 1 / ^ , /') > 0 . It remains only to consider the case ω = 0 and <p'(z) = (1 — ze~ iθή~ 2

THEOREM 3.2. Let feK(β) and let G(\ β) be as in Theorem 3.1. ( i ) If λ ^ 1, then lim inf (1 - r)^-%(r, f) ^ (ϋ) // λ ^ 1 and \{β + 1) > 1, then limβnp(1 - rγ^-Ur, f) ^ {β Note that when ω = 0, lim,..,! (1 — r) ;i(iS+1)~ 1/;(r, /) = 0, and when ω > 0 the growth of Iλ{r, f) is regular in the sense that lim sup^ and inf^i are either both positive or both zero.

LEMMA 4.1. Let g be given by (4.1). Then g is locally schlicht and vanishes only at the origin.

THEOREM 4.1. If feK0(β), then geB(l/β) where Conversely, if geB(ά), then fe KQ(l/a) where ι~ lla{g'{ξ)y ι«dξ.

Lemma 4.1 / i s regular in U, and since g e B(a) we have from the definition of / that where p e ^ and heS^*. Hence feKQ(l/a). Note that although for β > 1 / may be of arbitrarily high valence, it is always true that the corresponding g is schlicht. Also note that since Vk c K(k/2 — 1), we have a relation between Vk and B(2/(k — 2)). We now investigate the geometry of B(a). We shall assume that g is regular and locally schlicht in U, is normalized as in (1.1), and vanishes only at the origin. Al

THEOREM 4.2. With the above notation and hypothesis on g, we have that g e B(a) iff for all 0 < r < 1 the tangent to C(r) never turns back on itself as much as π radians.

COROLLARY 4.3. B(a) contains only schlicht functions.

Theorem 2.3. (See Theorems 8 and 9 of [4].). REFERENCES 1. I. E. Bazilevic, On a case of integrability in quadratures of the Loewner-Kufarev equation, Mat. Sborn., 37 (1955), 471-476. (Russian) 2. D. A. Brannan, On functions of bounded boundary rotation I, Proc. Edinburgh Mat. Soc, 16 (1968-69), 339-347. 3. A. W. Goodman, A note on the Noshiro-Warschawski theorem, (to appear). 4. , On close-to-convex functions of higher order, (to apper). 5. W. K. Hayman, The asymptotic behavior of p-valent func