ON CLOSED CONVEX HULLS AND THEIR EXTREME POINTS. S. K. Lee and S. M. Khairnar

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1 Kngweon-Kyungki Mth. Jour. 12 (2004), No. 2, pp ON CLOSED CONVE HULLS AND THEIR ETREME POINTS S. K. Lee nd S. M. Khirnr Abstrct. In this pper, the new subclss denoted by S p (α, β, ξ, γ) of p-vlent holomorphic functions hs been introduced nd investigte the severl properties of the clss S p (α, β, ξ, γ). In prticulr we hve obtined integrl representtion for mppings in the clss S p (α, β, ξ, γ) nd determined closed convex hulls nd their extreme points of the clss S p (α, β, ξ, γ). 1. Introduction Let S p (p fixed integer greter thn zero) denote the clss of functions of the form (1.1) z p + np+1 n z n which re holomorphic in the unit disc E {z : z < 1}. A function f in S p stisfying the inequlity ( Re z f ) (z) > α for z E nd 0 α < p, is known s p-vlent strlike function of order α. We denote it by S p (α). Now we define subclss S p (α, β, ξ, γ) of S p s follows. Received My 28, Revised October 22, Mthemtics Subject Clssifiction: 30C45. Key words nd phrses: p-vlent functions, holomorphic functions, extreme points, CLCO-closed convex hulls.

2 108 S. K. Lee nd S. M. Khirnr Definition.. A function f in S p belongs to the clss S p (α, β, ξ, γ) if f stisfies the condition. p ( ) ( ) 2ξ α γ p < β where 0 α < p, 0 < β 1, 1/2 < ξ 1, 0 < γ 1. By giving specific vlues to α, β, ξ, γ we obtin severl subclsses studied by vrious uthors Brickmn ([1]), Lkshminrsimhn([2]), Brickmn, Hllenbeck nd McGregor ([3]), Junej nd Mogr([4]). 2. Closed Convex Hulls nd Their Extreme Points To be ble to identify the closed-convex hulls nd the extreme points of closed convex hulls of S p (α, β, ξ, γ), we need the following lemm. Lemm. Let G(z) p + c 1 z + c 2 z 2 + be holomorphic in E nd stisfy the condition 2ξ(G(z) α) γ() < β for z E if nd only if G(z) 1 + βzθ(z)(2ξ γ) where 0 α < p, 0 < β 1, 1/2 < ξ 1, 0 < γ 1 nd θ(z) is holomorphic function such tht θ(z) 1. Proof. Suppose tht 2ξ(G(z) α) γ() < β for z E, where 0 α < p, 0 < β 1, 1/2 < ξ 1, 0 < γ 1. Let us consider the function (2.1) g(z) 2ξ(G(z) α) γ().

3 On Closed Convex Hulls nd Their Extreme Points 109 We observe tht g(0) 0 nd g(z) < β. By Schwrtz s Lemm we hve g(z) z. Thus we get (2.2) g(z) βzθ(z) where θ(z) is holomorphic in E nd θ(z) 1. From (2.1) nd (2.2) we hve Therefore βzθ(z) G(z) 2ξ(G(z) α) γ(). 1 + βzθ(z)(2ξ γ) where θ(z) is holomorphic in E nd θ(z) 1. Conversely, suppose tht G(z) hving bove representtion, is holomorphic in E. Then G(z) 1 + βzθ(z)(2ξ γ) where 0 α < p, 0 < β 1, 1/2 < ξ 1, 0 < γ 1 nd θ(z) is holomorphic in E nd θ(z) 1. Thus βzθ(z) 2ξ(G(z) α) γ(). Since θ(z) 1 nd z < 1, we hve βzθ(z) < β. Therefore 2ξ(G(z) α) γ() < β. Theorem 1. Let be holomorphic in E. Then S p (α, β, ξ, γ) if nd only if exp p βtθ(t)(γp 2ξα) (1 + βtθ(t)(2ξ γ))t dt where E \ 0. nd θ(t) is holomorphic in E nd θ(t) 1.

4 110 S. K. Lee nd S. M. Khirnr Proof. Let S p (α, β, ξ, γ). Then p ( ) ( ) 2ξ α γ p < β By Lemm or (2.3) f (z) 1 + βzθ(z)(2ξ γ) (1 + βzθ(z)(2ξ γ))z where θ(z) 1. Integrting both sides of (2.3), we hve log where E \ 0. Hence exp Conversely, ssume tht exp f (t) f(t) dt p βtθ(t)(γp 2ξα) (1 + βtθ(t)(2ξ γ))t dt p βtθ(t)(γp 2ξα) (1 + βtθ(t)(2ξ γ))t dt. p βtθ(t)(γp 2ξα) (1 + βtθ(t)(2ξ γ))t dt where 0 α < p, 0 < β 1, 1/2 < ξ 1, 0 < γ 1 nd E \ 0. Tking log nd differentiting both sides or By Lemm, 2ξ f (z) zf (z) Hence S p (α, β, ξ, γ). (1 + βzθ(z)(2ξ γ))z (1 + βzθ(z)(2ξ γ)). ( ) p α γ ) p < β. (

5 Remrk. On Closed Convex Hulls nd Their Extreme Points S p (α, 1, 1, 1) Sp(α) (H. Silvermn Clss). 2. S p (α, β, 1+α 2, 1) S p(β, 1+α 2 ) (T. V. Lkshminrsimhn Clss([2])). 3. S p (α, 1, β, 1) Sp(α, 1, β) (Junej Mogr Clss ([4])). Theorem 2. Suppose tht S p (α, β, ξ, γ). Then f hs the following integrl representtion ( ) 2ξ(γp α) z p exp log(1 + βxz(2ξ γ))dµ(x) 2ξ γ where is the unit circle nd dµ(x) 1. Proof. As S p (α, β, ξ, γ) we hve p ( ) ( ) 2ξ α γ p < β where 0 α < p, 0 < β 1, 1/2 < ξ 1, 0 < γ 1. By routine clcultion, we hve z f (z) p βxz(γp 2ξα) 1 + βxz(2ξ γ) dµ(x), where is the unit circle nd dµ(x) 1. This cn be further trnscribed s, ( ) 2ξ(γp α) log z p log(1 + βxz(2ξ γ))dµ(x). 2ξ γ This is equivlent to ( 2ξ(γp α) z p exp 2ξ γ ) log(1 + βxz(2ξ γ))dµ(x). We now stte very generl result tht is originlly due to Brickmn, Hllenbeck. Our findings hevily depend on this very useful result. This result is populrly described s Geometric Men Theorem.

6 112 S. K. Lee nd S. M. Khirnr Theorem 3. Let be the unit circle, P the set of probbility mesures on nd p > 0. Then given µ P, there exists γ P such tht ( ) exp p log(1 xz)dµ(x) (1 xz) p dγ(x). We shll put to use the integrl representtion of the members of S p (α, β, ξ, γ) long with the Geometric Men Theorem to determine the set of extreme points of the closed convex hull of S p (α, β, ξ, γ). Theorem 4. Let be the unit circle, P the set of probbility mesures on nd F the set of functions f µ on E, defined by f µ (z) z p (1 + βxz(2ξ γ)) 2ξ(γp α) 2ξ γ dµ(x) where µ Pnd 0 α < p, 0 < β 1, 1 2 < ξ 1, 0 < γ 1. Then F CLCO S p (α, β, ξ, γ) nd the mp µ f µ is one to one nd ech function is in the Ext. CLCO S p (α, β, ξ, γ). z z p (1 + βxz(2ξ γ)) 2ξ(γp α) 2ξ γ Proof. By theorem 2 nd 3, clerly if f S p (α, β, ξ, γ), then f F is compct nd convex. Hence F is the closed convex hull of P. Thus CLCOS p (α, β, ξ, γ) F. Agin s ech kernel function z z p (1 + βxz(2ξ γ)) 2ξ(γp α) (2ξ γ) belongs to S p (α, β, ξ, γ). Thus F CLCO S p (α, β, ξ, γ). Therefore F CLCO S p (α, β, ξ, γ).

7 On Closed Convex Hulls nd Their Extreme Points 113 If f µ1 f µ2 for µ 1 P nd µ 2 P, then z (1 + βxz(2ξ γ)) 2ξ(γp α) (2ξ γ) z dµ 1 (x) (1 + βxz(2ξ γ)) 2ξ(γp α) (2ξ γ) for z E. It follows tht x n dµ 1 (x) x n dµ 2 (x) dµ 2 (x) for n 0, 1, 2,. Therefore we conclude tht µ 1 µ 2. Hence the mp µ f µ is one-to-one nd we get the desired conclusion bout the extreme points. We now stte severl prticulr cses of our result. The very fct our result subsumes severl prticulr cses mny of which seem to be new stnds in testimony of the effectiveness nd utility of the new fmily of S p (α, β, ξ, γ) tht we hve introduced. Corollry 1. Let be the unit circle, P the set of probbility mesures on nd F the set of functions f µ on E defined by f µ (z) z(1 + xz) 2 dµ(x) where µ P. Then F CLCO S(0, 1, 1, 1) CLCO S nd the mp µ f µ is one-to-one. This corollry is due to ([1], p1). By giving specific vlues sy α 0, β ξ γ 1 in Theorem 4, we get the fmily S of univlent strlike functions of order α. Corollry 2. Let be the unit circle, P the set of probbility mesures on nd F the set of functions f µ on E defined by f µ (z) z(1 + αβxz) 1+α α dµ(x)

8 114 S. K. Lee nd S. M. Khirnr where µ P, 0 < α < 1, 0 β < 1. Then F CLCO S(α, β, 1 + α 2, 1) CLCO S (β, 1 + α ). 2 Further the mp µ f µ is one-to-one nd ech function z z(1 + αβxz) 1+α α is in the Ext. CLCO S(α, β, 1+α 2, 1). This corollry is due to ([2], p1). By giving specific vlues sy α α, β β, ξ 1+α 2, γ 1 in Theorem 4, we get the fmily S (β, 1+α 2 ) of univlent strlike functions of order α. Corollry 3. Let be the unit circle, P the set of probbility mesures on nd F the set of functions f µ on E defined by, f µ (z) where µ P, 0 α < 1. Then z(1 + xz) 2(1 α) dµ(x) F CLCO S(α, 1, 1, 1) CLCO S (α). Further the mp µ f µ is one-to-one nd ech function is in the Ext. CLCO S(α, 1, 1, 1). z z(1 + xz) 2(1 α) This corollry is due to ([3], p1). By giving specific vlues sy α α, β ξ γ 1 in Theorem 4, we get the fmily S (α) of univlent strlike functions of order α. Corollry 4. Let be the unit circle, P the set of probbility mesures on nd F the set of functions f µ on E defined by f µ (z) z(1 + xz(2β 1)) 2β(1 α) 2β 1

9 On Closed Convex Hulls nd Their Extreme Points 115 where µ P, 1 2 < β 1, 0 α < 1 2β. Then F CLCO S(α, 1, β, 1) CLCO S (α, β). Further the mp µ f µ is one-to-one nd ech function is in the Ext. CLCO S(α, 1, β, 1). z z (1 + xz(2β 1)) 2β(1 α) 2β 1 This corollry is due to ([4], p1). By giving specific vlues sy α α, β 1, ξ β, γ 1 in Theorem, we get the fmily S (α, β) of univlent strlike functions of order α. References 1. Brickmn L., McGregor T. H. nd Wilken D. R., Convex hulls of some clssicl fmilies of univlent functions, Trns. Amer. Mth. Soc. 156 (1971), Lkshminrsinhn T. V., On subclsses of functions strlike in the unit disk, Jour. Indin Mth. Soc. 41 (1977), Brickmn L., Hllenbeck D., McGregor T. H nd Wilken D. R., Convex hulls nd extreme points of fmilies of strlike convex mppings, Trns. Amer. Mth. Soc. 185 (1973), Junej O. P. nd Mogr M. L., Rdii of convexity for certin clsses of univlent nlytic functions, Pcif. Jour. Mth. 78 (1978), S. K. Lee Deprtment of Mthemtics Eduction Gyeongsng Ntionl University Chinju, , Kore E-mil: sklee@gsnu.c.kr