V28N1-P17-32_OCR

Theorem 2 .1. If (X) L(2n - 1 - a)nlanl :S 1 - a n=2 (2.3)

Theorem 2.2. Let f(z) = z- I::=Z anzn, an~ 0 then I::=2 n(2n-1-a)an::::; 1-a if and only if f(z) is in UCT(a).

Corollary 2.1. UCT(a) = C*(11et).

Corollary 2.2. If f(z) E UCT(a), then 1-a 2 1-a 2 r- 2(3:-a{ :S lf(z)I :Sr+ 2(3-a.r 1-a 1-a 1- --r :S !f'(z)! :S 1 + --r 3-a 3-a and the extreme points are 1-a fi(z)=z, fn(z)=z- ( \zn,

Theorem 2.3. A function f(z) = z - L:=2 anzn is in UCT(a, /3) if and only if 00 I)n(l + a) - (a+ /3)]nan :S 1 - /3. n=2 (2.5)

Corollary 2.3. UCT(a,(3) = C*(~:!).

Corollary 2.4. If f(z) E UCT(a, (3), then l-(3 2 l-(3 2 r- _,_ f3{ :S j/(z)j :Sr+_,_ _,r l-(3 1-(3 1- 8r::;jj'(z)j:Sl+ r 2+a- 2+a-(3

Theorem 2.4. If I::=2(2n - 1 - a)lanl ::::; 1 - a then f(z) = z + I::=2 anzn is m Sp(a).

Theorem 2.5. f(z) = z- I::=2 anzn, an~ 0 is in SpT(a) if and only if I::=2(2n- 1 - a )an ::::; 1 - a.

Theorem 2.6. f(z) = z - I::=zanzn, an ~ 0 is in SpT(a,,B) if and only if I::=2[n(l + a) - (a+ ,B)]an ::::; 1 - ,8.

Corollary 2.5. If f(z) E SpT(a), then 1-o: 1-o: r - --r2 :S lf(z)I ::::; r + --r2 3-a 3-a 1- 2(1-a)r::::; lf'(z)I::::; 1+ 2(1-a)r 3-a 3-a and the extreme points are 1-a n Ji ( z) = z and f n ( z) = z - 2n _ 1 - a z ' n = 2,3,···.

Corollary 2.6. If f(z) E SpT(a,,B) then 1-,8 1-,8 r - r2 < If (z) I < r + r2 2+a-,B - - 2+a-,B 1+ 2(l-,B) r<jf'(z)l<l+ 2(l-,B) r 2+a-,B - - 2+a-,B

Theorem 2.7. If 00 L[n + 2(a - l)]nlan! :S 2a - 1 n=2 (2.6) then f(z) of the form (i) is in CP(a).

Theorem 2.8. f(z) = z - I:::=2 anzn, an 2 0 is in CPT(a) if and only if 00 L n( n ~ I + a )an :S a. n=2 (2.8)

Corollary 2.7. CPT(a) = C*(l - a) for O < a :S 1.

Corollary 2.8. If f(z) E CPT(a), then for O < a :S 1 r - _ , . a , r2 :S If ( z) I :S r + _ , . a , r2 1- _a_r < lf'(z)I < 1 + _a_r l+a - - l+a and the extreme points are a n fi(z)=z andfn(z)=z- n(n-I+a\z' n = 2,3, ·· ·.

Theorem 2.9. f(z) = z - I::=2 anzn, an ~ 0 is in PT(a) if and only if 00 L) n + a - 1 )an ::::; a. n=2 (2.9)

Corollary 2.9. PT(a) = T*(l - a), 0 <a::::; 1.

Corollary 2.10. Let f(z) E PT(a). Then for O <a::::; 1 a 2 a r - --r < lf(z)I < r + --r2 l+a - - l+a 2a , 2a 1 - --r < If (z)I < 1 + --r l+a - - l+a

Theorem 3.1. If f(z) = z - :z=:=Z anzn, an 2 0 and g(z) = z - L:=2 bnzn, bn 2". 0 are elements of SpT(a), then (f * g)(z) = h(z) = z - L:=2 anbnzn is in SpT(/3) where /3 = /3(a) = 2(2~~)' 0 :Sa< 1. The result is best possible.

Corollary 3.1. For f(z) and g(z) as in Theorem 3.1 we have 00 h(z) = z - L va::,;;;, n=2

Theorem 3.2. For f(z) E SpT(a) and g(z) E SpT(f3) we have 3 - a(] f(z) * g(z) E SpT(,i -· ~) = SpT(,). The result is sharp.

Corollary 3.2. Let f(z), g(z) and h(z) E SpT(a). Then f(z) * g(z) * h(z) E SpT(/3) where {3 = 12-9et+et~ . 13-12et+et2

Theorem 3.3. Let f(z) = z- I::=2 anzn, an ~ 0 and g(z) = z- I::=2 bnzn, bn ~ 0 be elements of UCT(a) then f(z) * g(z) = h(z) = z - I::=2 anbnzn E UCT(/3) where {3 = {3( a) ::::; 2(~~C;r~_); . The result is sharp.

Theorem 3.4. Let f(z) E UCT(a) and g(z) E UCT(/3) then f(z) * g(z) E UCT(,) _ ( /3) 24-(3+a)(3+/3) 1 - 1 a, ' :S (5 - a)(5 - /3) - 8 · The result is sharp.

Theorem 3.5. Let f(z) = z- L~2 anzn, an 2 0 and g(z) = Z- L:=2 bnzn, bn 2". 0 be in SpT(a, /3). Then (f * g)(z) = h(z) = z - L:=2 anbnzn E SpT(A, B) where a+ /3 A= A(a, /3) = 2[1- --] l+a a-/3 a+/3 B = B(a,/3) = --[2- --]. l+a l+a

Definition 1.1.([1]) A function f(z) is uniformly convex in E if f(z) is in Kand has the property that for every circular arc I contained in E, with centre c also in E, the arc f(,) is convex. Ronning [2] found a more applicable one variable analytic characterization for the above class denoted by UCV. Definition 1.2.([2]) A function f(z) = z + I::=2 anzn is in UCV if and only if zf"(z) I zf"(z) I Re 1 + f, ( z) ~ f, ( z) ' z E E.

Definition 1.3.([5]) Let T be the subfamily of S consisting of functions of the form 00 f(z) = Z - L anzn, n=2 an 2". 0, z EE. (1.2) Let T* (a) and C* (a) be the subfamily of functions in T that are starlike of order a and convex of order a respectively for O :S a < 1. Silverman [5] gave a co-efficient characterization for these classes.

Definition 1.4.([4]) A function f(z) is in Sp(a) if f(z) satisfies the analytic charac- terization l zf'(z) / R zf'(z) f(z) - 1 :S e f(z) - a and f(z) is in UCV(a) if and only if zf'(z) is in Sp(a). (1.7)

Definition 1.5. ([4]) A function f(z) = z+ I::=2 anzn is in the class P(a,(3) if f(z) satisfies the analytic characterization I z f' ( z) - ( a + (3) I < Re z f' ( z) + a - f3 f(z) - f(z) 0 < a < oo, 0 :S (3 < l, z EE. This means that zJ~~)) for f(z) E p(a,(3) and z EE lies in that portion of the plane which contains w = l and is bounded by the parabola y2 = 4a(x - (3). We observe that

Definition 2.1. Let UCT(a) be the class of functions f(z) = z-I:::=2anzn, an 2 0 satisfy the condition ' zf"(z)' zf"(z) f'(z) :SRe{l+ ;1f,,\ -a}, z EE. (2.1) Definition 2.2. Let SpT(a) be the class of functions f(z) = z-I:::=2anzn, an 2 0

Definition 2.3. Let UCT(a,/3) be the class of functions f(z) = z - L:=2 anzn, an > 0 that satisfy the condition zfu(z) /zfu(z)/ Re{ 1 + f'(z) } ~ a f'(z) + /3, We write UCT(l, /3) = UCT(/3) and observe that UCT(O, /3) = C*(/3). a 2". 0, /3 2". 0.

Definition 2.4. For f(z) = z + :z=:=2 anzn, let z E f, 0 <a< oo }.

Definition 2.5. Let CPT(a) be the class of functions f(z) = z- :z=:=2 anzn, an 2". 0 such that f(z) is in CP(a). Since f(z) E CPT(a) {=> zf'(z) E PT(a), we get that l zf'(z) / zf'(z) PT (a) = { f ( z) E T; f( z) - a :S Re £ 1 _ \ + a, z EE, 0 <a< oo }.