DEFINED BY SUBORDINATION

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 1, October 1974 EXTREME POINTS OF CLASSES OF FUNCTIONS DEFINED BY SUBORDINATION D. J. HALLENBECKl ABSTRACT. The closed convex hull and extreme points of families of analytic functions, which are defined in terms of subordination, are determined. Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. These results are used to solve extremal problems. The functions we consider are defined by subordination to certain classes of starlike and convex mappings. 1. Introduction. In this paper, we shall determine the extreme points and the convex hulls of families of analytic functions, which are defined in terms of subordination. In [5], T. H. MacGregor and this author initiated this study. We shall extend some results of that paper and give some applications of new results. Let A denote the unit disk. We say / is subordinate to F, written/ -< F, if there exists an analytic function <p, known as a Schwarz function, so that 0(0)= 0, <tso) < 1 and f(z) = F(cp(z)). We let St(a), where a< 1, denote the set of functions which are starlike of order a. We recall that f(z) St(a) if and only if /(0) = 0, /'(0) = 1 and ReO/'0)//0)) > a for z in A. We let K(a), where a< 1, denote the set of functions which are convex of order a. We recall that f(z) K(a) if and only if /(0) = 0, /'(0) = 1, and Re(l + zf"(z)/f'(z)) > a for z in A. We note that St(a) and K(a) are both compact families of analytic functions. We let St (a.) = {/: / -< g tor some g in St(cx)! and K*(a) = }/:/-< g for some g in K(a)\. In [6, p. 365], T. H. Mac- Gregor proved that for any compact family F of analytic functions F = {/:/-< g for some g in F i is a compact family. Hence St (a) and K (a) ate compact. We let S djnote the set of analytic functions on A which are univalent and satisfy /(0) = 0, /'(0)= 1. It is known that St(a) C S for 0< a< 1 Received by the editors July 26, AMS (MOS) subject classifications (1970). Primary 30A32. Key words and phrases. Starlike functions, convex functions, close-to-convex function, extreme point, probability measures, integral representation, subordination. 1 This research was, in part, supported by a University of Delaware Faculty Research Summer Grant. 59 Copyright 1974, American Mathematical Society

2 60 D. J. HALLENBECK and K(a) C S for - la < a < 1- For any compact family of analytic functions F we let S\F denote the closed convex hull of F and fcjlf denote the extreme points of the closed convex hull We shall determine KSt (a) for a= % and a< 0. We also determine K7<*(a)for a< Extreme points and convex hull of K (a). Theorem 1. Let X x Y = i(x, y): x = \y\ = l}, 3* be the set of probity measures ability defined by on X x Y, 0-< 0, czw J be the set of functions f on A then Kk (a) = J and &KK*ia) = (1-2a)-1xy-1((l - yz)~ (1 "2a) - 1): x = y = 1 Proof. Suppose / ëh/( (a); then, as was pointed out in [6, p. 366], f < g for some function g e ëhk(a). In Theorem 4 [2] the set &KK(a) was proved to be the set of functions g(z) = (l-2a)-1y-1((l-y^)-(1-2a)-l) for \y\ = 1. So /U) = (1-2a)-'y" All - y</»u))-(1-2a)- D for some y satisfying y = 1 and some Schwarz function </3(z). Let Kiy)=\f: f<g, g(2) = (l-2a)-1y-1((l-y^)-(1-2a)- 1)1. Since K(y)C fc*(a) we conclude that JÍK(y)C HK*(a) and / ejík(y). Let tf = {A: A -< (1 -z)-(1-2a)}., The map h -* (1-2a)-1y"I(M»)- Ù is a linear map from 77 to K(y) and consequently from &Hf7 to &KK(y). We recall that in Theorem 2.2 [l ] ëmr7 was determined to be the set {(1 - xz)-(1-2a': x = 1} since a< 0 implies that 1-2a> 1. Hence we see that f(z)=il -2a)-1y-1((l-xz)-(1-2a)-l)=(l-2a)-1x(yx)-I((l-x2:)-(1-2^-l) which by a renaming of letters becomes (l-2a)-ixy-1((l-y2)-(i-2a)-d.

3 EXTREME POINTS OF CLASSES OF FUNCTIONS 61 Hence &KK*(a)C F = (l-2a)-1xy-1((l-yz)-(i-2a)- 1): x = \y\ = If. We prove F C &Kk (a) by showing that each function in E uniquely maximizes a real-valued continuous linear functional over F. We note that /(z) = (l-2a)-1xy-1((l-yz)-(i-2a)-l) is in K (a) for each pair x, y satisfying x = \y\ = 1. Let ](/)= af'(0)+ bf"(0) where a = x~ and b = (xy)~ for a fixed pair x = \y\ = 1. For any f(z) E we see that Re ](f)< 3-2aand that Re /(/)= 3-2a if and only if x = a-1 and xy = b~l. Hence, for the choice of a and b indicated above Re J(f)= 3 2cl So a unique function in F maximizes Re J(f) over F for each pair x = y = 1. Hence F C &H.K (a). The conclusion of the theorem now follows from the Krein-Milman theorem [4, p. 440] and Theorem 1 in [3, p. 93]. determined. Remarks. 1. This result generalizes Theorem 1 in [5] where JlK is 2. We recall that K(ol) C when - V2<a< 0 and denotes the class of close-to-convex functions. By applying the preceding theorem, we see that ßKKA-W^lz-Wxz^l-xz)-2: * = 1}. We next introduce some notation. Suppose f(z) = _0<z zn and F(z) = 2 na z" are analytic functions in A. Suppose \a I < A I for 72 = 0, 1, 2, 72 = 0 n ' rr ' n ' ' nl ' ' '. We denote this relation by / «F. 7 1 Theorem 2. // /O) -< F(z), F(z) K(a), and a < 0, ' e72 /(z) «(1-2a)-J((l - 2)-(1-2a) - 1). Proof. Since it suffices to consider as candidates for F(z) the functions in tejlk (a), the result follows immediately from the previous theorem. 3. Extreme points and convex hull of St (a). Theorem 3. Let X x Y and J be as in Theorem 1, a= V2 or a< 0, and j be the set of functions f analytic on A defined by Then &t*(a) ='S and &ist*(a) = {yz{\ - xz)-{2-2a): \x\ = \y\ =1. Proof. Suppose a < 0 and / fekst (a), then as remarked earlier we have / -< g for some function g in ëhst(a), But in [2, Theorem 3] the set

4 62 D. J. HALLENBECK ëhst(a) was determined to be the set of functions \g(z) = z(l - xz)~ " ' : \x\ = 1} Hence f(z) = (f>(z)(l - xcb(z))~^ ~ a for some x satisfying x = 1 and some Schwarz function (f>(z). Let A=\f:f< z(l - xz)~kl~l ', x = 1}. It is clear that yz(l -*z)-(2-2a> for each pair x = \y\ = 1, since yz(l - xz)~(2~2a) = (f>(z)(l - xyt/.(z))-(2-2a) e A where cb(z) = yz. Hence ï C JÍA. We next show that AC?. Suppose f(z)< z(l -xz)-{2~2a) for some x = 1. Hence fiz) =-- -rrrrsr7~ T\ = / n \i'fc «f T^~d^ "> (1 - x< U))1 ^ 1 - x<p(z) JX(1 - xz)1_2a JyxVfl-wz 7 where X = \x: x = 1}, Y = \y. \y\ = 1} and W = \w: \w\ = 1}. This follows Theorem 1 of [5] and Theorem 2.2 of [l] since 1-2a> 1. Hence we have by the Fubini theorem J. -~=-daix) dviy, w). Xx(YxWy d - xz)1-2^! - wz) * y It suffices to prove yz(l xz)~ ~ (1 wz)~ J" since J" is closed and convex. However, by Theorem 1 of [2] we have K(i _yz_f_m_-ma a-xz)1-2aa-wz) hd-tz)2- U - xz) vi - WZ) ' 1 \L - IZ) dnit) where T = \t: \t\ = 1} and n(t) is a probability measure on T. Hence, it suffices to prove that yz(l - tz)~^ ~ a' is in 3" for each pair \y\ = \t\ = 1 This is clearly true and hence A C 3" and so KA C K? = J. As remarked above, we know that ëhst (a) C A. Hence KSt (a) = HëKSt (a) C KA = Jr by the Krein-Milman theorem since KSt (a) is compact and convex. Since yz(l - xz)~ ~ = </>(z)(l xy~cf>(z))~^ ' ' where r/j(z) = yz we conclude that yz(l - xz)~ ~ is in St (a) for each pair x = \y\ = 1 Hence if C Kst (a) by Theorem 1 [3] and consequently HSt (a) = 3". Since HSt*(a) = 5F we have êhst*(a)c E where E = \yz(l - xz)~(2-2a): \x\ = \y\ = l\, To prove fejlst (a) = E we need only show that each function in E uniquely maximizes a real-valued continuous linear functional over This follows as earlier in part because E C St (a). Let /(g) = ug (0)+ vg (0) where \u\ = \v\ = 1. We then have by Re /(g) =ReUy + 2(2-2a)Vyz\ < \Uv\ +4(1 - o)\vyz\ = 5-4a.

5 EXTREME POINTS OF CLASSES OF FUNCTIONS 63 The maximum value of 5-4a is only achieved if y = u~ and x = u/v. Letting u and v vary, we get all possible pairs x, y associated with the unique function in E with Re ](g) = 5-4a. Hence F C tehst (a) and consequently 5HSt*(a) = E. Remarks. 1. Theorem 3 generalizes Theorem 2 in [5] by the present author and T, H. MacGregor, 2. We recall that Theorem 3 in [2] when a = ]/2 and k = 1 showed that ëhst(h) = &Kk where the latter set was determined in Theorem 2 in [3, p. 94]. This fact, coupled with an examination of the proof in Theorem 1 of this paper when a = 0, shows that Theorem 3 above holds when a= l/2. 3. It was proven in Theorem 7 in [2] that if / -< F where F is in St(a) and ]/2< a< 1 then f(z) «z/(l - z). Hence Theorem 3 cannot hold when Y2 < a < 1. So KSt*(a) t? when ]/2 < a < 1. We conjecture that Hst*(a) = Î when 0 < a < ]/2 but we have been unable to prove this. 4. Theorem 3 affords a quick proof of Theorem 7 in [2]. If / -< F and F e St(a)when a< 0 then F(z)<< z(l - z)-(2_2a)and if a= ^ then /0)«z/(l -z). 4. A coefficient estimate for functions in St (a) when 0< a< l/2 Theorem 4. // f(z) = S^, «nzn < FO) ^Aere FO) e St(a) and 0 < a < % then \a I < (1-2a)?2 + 2a = ß(a) ' Proof. It suffices to consider F(z)= z/(\ z).we have <j>(z) 1 /U I-cMzHI-ctSU))1-2*' We let 0O)/(1-00))= bxz + b2zz +... and (1-0O)rU-2a,= l + Cjz + 2 c.z +. We have a =& + >,c,+b ^c, + + >,c, and we know that \b < 1 for 72 = 1, 2, by Theorem 1 when a = 0 or by the classical result of Rogosinski [7]., We also know that \c < 1 2cx for n = 1, 2, by another theorem of Rogosinski [7] with equality never holding for two values of 72 simultaneously. Hence by the triangle inequality \a I < (1-2a)72 + 2a. Equality is not possible since \c \ cannot equal 1-2a for more than a single value of 72. Remarks. We note that S(0)= 72, the classical result of Rogosinski [7] and B(lA) = 1 which is contained in Theorem 7 in ul We conjecture that f(z) «z/'l - z)2~ a which is known to hold for a< 0 also holds for 0 < a < y2.

6 64 D. J. HALLENBECK REFERENCES 1. D. A. Brannan, J. G. Çlunie and W. E. Kirwan, On the coefficient problem for functions of bounded boundary rotation, Ann. Acad. Sei. Fenn. AI 523 (1973). 2. L. Brickman, D. J. Hallenbeck, T. H. MacGregor and D. R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 (1972), L. Brickman, T. H. MacGregor and D. R. Wilken, Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc. 156 (1971), MR 43 # N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, MR 22 # D. J. Hallenbeck and T. H. MacGregor, Subordination and extreme point theory, Pacific J. Math, (to appear). 6. T. H. MacGregor, Applications of extreme point theory to univalent functions, Michigan Math. J. 19 (1972), W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (2) 48 (1943), MR 5, 36. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF DELAWARE, NEWARK, DELA- WARE 19711