The Multivalent Class of Geometrically Close-to-Convex Functions

THEOREM 3.1. Let F be a nonconstant function analytic in B with F(0) = 0. The following conditions are equivalent: (A) F' has p — 1 zeros in B, and F e Kw(p). (B) F' hasp — 1 zeros in B, and there is f e Sw(p), with /(0) = 0, such that lim inf[min Re(zF7/) ] ^ 0. r-*r \z\=r (C) JF /jos p — 1 zeros /'« B, and there are functions Fn G K(p) and f G Sa(p), withffl(0) = 0 /or all n, such that Fn —> F and fn -* f locally uniformly in B, / e Sw(p)> and Re(zFn'/fi) > 0 /or a// z, 0 < pn < |z| < 1.

LEMMA 3.3. Let P be a polynomial and D be a domain. Suppose that j or any two points in dD we can find a P-ray or two disjoint P-rays containing the points and not meeting D. Then for any n points Zj, z2, . . . , zn in dD there exist a finite number of mutually disjoint P-rays containing the points and not meeting D.

Lemma 3.3 and the proof of Theorem 4.1 in [8], for every F there is a https://doi.org/10.4153/CJM-1987-013-0 Published online by Cambridge University Press

Theorem 2.8 in [9] that there exists a pair of P, <j> where P is a polynomial of degree/? and <$> e S, such that C — <J>(B) is a union of a collection, W, of P-rays with the properties: Each ray starts from the boundary of <£(B), and for any two rays either they have disjoint interiors or one is a subset of the other. We construct via W a ruling, ££, of C — <J>(B) consisting of P-rays which start from 3^>(B) and have mutually disjoint interiors. Let-^o be the collection of all P-rays or lines, /$

COROLLARY 4.1. Bf(p) is a subset of S. This is a special case of a more general result due to Bazilevic [1] (see also [11]).

COROLLARY 4.2. A univalent function <j> belongs to B\p) if and only ij C — <KB) is a union of zp-rays of disjoint interiors. https://doi.org/10.4153/CJM-1987-013-0 Published online by Cambridge University Press

LEMMA 4.1. Let a be a positive real number, and let m be a positive integer. Suppose that f e S. Let g be the m-fold symmetric function of f that is, g(z) = [f(zm)]u">. The C — /(B) is a union of za m-rays of disjoint interiors if and only if C — g(B) is a union of za-rays of disjoint interiors.

LEMMA 4.2. Under the assumptions of the above lemma, the function g belongs to B\a) if and only if f belongs to B\a/m). Now we have:

Definition 2.1. Let S be the familiar class of functions /univalent in B that satisfy /(0) = 0 and /'(O) = 1. Received December 6, 1984 and in revised form November 6, 1985. 297 https://doi.org/10.4153/CJM-1987-013-0 Published online by Cambridge University Press

Definition 2,2. Let S* be the class of functions / t h a t satisfy one of the following conditions: (a) fis univalent in B, /(0) = 0, and /(B) is starshaped with respect to the origin. (b) / is analytic in B, admits one zero there (counting multiplicity), and Re(z/7/) > 0 for all z e B Note that the normalization f'(0) = 1 is not required in this

Definition 2.3. A function / i s said to belong to Sa(p) if / i s analytic in B, has p zeros there, and there is an annulus Ap = {z:p < \z\ < 1} such that Re(z/7/) > 0 for all z e Ap. Let *(z, 0 = (z - 0 0 - Sz)/z, and let *(z, 0) = 1. Hummel [3] has extended the class Sa(p) to weakly starlike functions of order p as follows:

Definition 2.4. A function / i s said to belong to Sw(p) if / i s analytic in B, has p zeros there, and satisfies one of the conditions: (a) lim inf [min Re(z/7/) ] ^ 0. (b) There exists a sequence (fn)™=\ where fn e Sa(p) for all «, such that / —>/ locally uniformly in B. (c) There is h G S* such that /(z) = \h{z)rn*(z,z,\ iz,.i< i,i ^/-^/7. / = i Observe that this class is not closed in the topology of locally uniform

Definition 2.5. A function / i s said to belong to Swc(p) if / i s analytic in B and satisfies one of the conditions: https://doi.org/10.4153/CJM-1987-013-0 Published online by Cambridge University Press

Definition 2.6. A function F belongs to K(p) if F is analytic in B, F(0) = 0, and F satisfies one of the conditions: (a) There is / e Sa(p), with /(0) = 0, and an annulus Ap such that Re(zF'//) > 0 for all z G ^p. (b) Ff has/? — 1 zeros in B, and for any 6X < 62 and p < r < 1 J I Re(l + reieF"{rëe)/F\rëe))dQ > -77. Note that i^(l) = ,K, and that AT(/?) is not closed in the topology of locally uniform convergence in B. This has led to the following extension, the class of weakly close-to-convex f

Definition 2.7. Let F be a nonconstant function analytic in B with F(0) = 0. F is said to belong to Kw(p) if one of the following conditions is satisfied: (a) There is / <= Sw(p\ with /(0) = 0, such that lim inf[min Re(zF'/f) ] ^ 0. / — I " \z\=r (b) There are functions Fn e ^(/?) and fn e Sa(p), with each / ( 0 ) = 0, such that Fn -^ F and / —»/, locally uniformly in B, / e Sw(p),

Definition 2.7. Conditions (H), (J), (K), and (L) are suggested by [8] and [9]. Condition (I) is an extention of Kaplan's criteria for the class K, and later Livingston's criteria for the class K(p) (see [4] and [7] ). Condi- tion (M) is motivated by the work of Sheil-Small on linear accessibility (see [14] ). LEMMA 3.1. Let 6 = 0, and let Tbe a real-valued function on( — oo, oo) that satisfies (a)

Definition 2.7 (e) there exist / e Swc(p), with f(0) = 0, and a function h of. positive real part such that zF' = fh in B. Since Ff has exactly p — 1 zeros in B, / h a s exactly p zeros in B. Hence / e Sw(p) a n d F satisfies condition (F). Conversely, (F) => (A) follows by reversing the previous argument. (A) <=> (G). This is straightforward from Definition 2.7 (c) and the argument principle. (G) => (H). Suppose F satisfies condition (G). Then there are functions Fn G K(p) and p, 0 < p < 1, suc

Definition 3.1. Let K (p) be the class of functions F analytic in B, with F(0) = 0, such that F' has exactly p — 1 zeros in B, and F satisfies one of the conditions (A), (B), . . . , (M). We call Kg(p) the class of geometrically close-to-convex functions of order p. 4. K (p) and Bazilevic functions. We deal here with the special class, B'(a), of Bazilevic functions of order a. Definition. 4.1. Let B\a), 0 < a < oo, be the class of all functions where /z(f) = £ + . . . e S*, and g(f) = 1 + a,f +

Definition 4.2. Let Bg(p) be the class of all functions <J> e S such that Po(f) G Kg(p) for some polynomial P of degree/?. We relate the classes B\p) and Bip) as follows: THEOREM 4.1. (a) 5'(1) = Bg(\) = K. (b) For p ^ 2, P'(p) Is a proper subset of Bip).