V26N3-P201-223_OCR

Lemma 1. If a function 00 H(z) = 1 + L dnzn. n=l (2.1) is analytic in U and satisfies the condition I H(z) - 1 I 1 (B - A)/3(H(z) - 1 + (1 - a)e-i>. cos>..)+ A(H(z) - 1) < ' (2.2) for i>..I < ~' 0 :s; a< 1, 0 < /3 :s; 1, -1 :s; A< B :s; 1, 0 < B :s; 1, and for all z EU, then

Theorem 1. A Junction J(z), defined by (1.1) and analytic in U, is in the class s>..(a, ,B, A, B) if and only if f(z) =z exp {-(B - A)/l(I - a)e_,, cos,\ f.' I+ [(B _ :\~ + A]t,p(t) dt}. (z EU) (2.6) for some <p(z) E Q.

Corollary 1. Let the function f(z) defined by (1.1). Then f(z) E S\a, /3, A, B) if and only if there is a function fi(z) E S*(a,/3,A,B) such that e-i.>. cos A f(z) = z [!i;z)l . (z EU). (2.10) In view of the relationship (1.6), it is not difficult to deduce from the above results the following representation formulas for functions belonging to the class CA(a, /3, A, B):

Corollary 2. A function f(z) defined by (1.1) is in .the class CA(a, /3, A, B) if and only if its derivative f'(z) can be represented as follows: (i) J'(z) = [J~(z)ie_·.,. cos A for h(z) E C0(a,/3,A,B) = C*(a,/3,A,B); ( ii) (2.11) . { . r cp(t) dt } (2 12) J'(z) = exp -(B - A)/3(1 - a)e-iA cos.-\ Jo 1 + [(B - A)/3 + A]tcp(t)

Theorem 2. Let the function f(z) defined by (1.1) be analytic in U. Then f(z) E SA(a, /3,-A, B) if, for some .X, a, A and B (I.-\I < i; 0 :s; a< 1; -1 :s; A < B :s; 1; 0 < B :s; 1), 00 L {n[l - A - (B - A)/3] - 1 + 1(-A - (B - A)/3] + (B - A)/3(1 - a)e-iA cos .-\I} lanl n=2 :s; (B - A)/3(1 - a) cos .-\, (3.1) whenever O < /3 :s; ( A~B ), and 00 L { (n - 1) + I [(B - A)/3 + A](n - 1) + (B - A)/3(1 - a)e-iA cos ..Xj} Ian I n=2 :s; (B - A)/3(1 - a) cos .X, (3.2)

Corollary 3. Let the function f(z) defined by (1.1) be analytic in U. Then f(z) is in the class c>.(a,/3,A,B) if, for some i, o:, A and B(l>..I < ~; 0 :s; o: < 1; -1 :s; A< B :s; 1; 0 < B :s; 1). (3.6) 00 L n { n[l - A - (B - A)/3] - 1 + ![-A - (B - A)/3] + (B - A)/3(1 - o:)e-i>. cos >..I} lanl n=2 :s; (B - A)/3(1 - a) cos >.., (3.7)

Theorem 3. If a function f(z) defined by (1.1) is in the class s>..(a, ,B, A, B), then lf(z)I::; r [(l + [(B - A),B + A]r)(l-cos >..)] <B;,11f3<~>;~~'\"" (1 - [(B - A),B + A]r)(l+cos >..) ( 4.1) and [ (1 [(B A) f.? ( (B-A)/3(1-a) cos A lf(z)I ~ r - - JJ + A]r) 1-cos >..)]

Corollary 4. If a function f(z) defined by (1.1) is in the class c>.(o:,/3,A,B), then (4.10) [ (1 + [(B A)/3 ( (B-A)/3(1-<z) COH;,.. IJ'(z)I ~ - + A]r) 1-cos ,\)] 2[(B-A)f1+AJ 1 - [(B - A)/3 + A Jr )(l+cos >.) ( 4.11) and [

Lemma 2 [24]. Let the function w(z) defined by 00 w(z) = L CnZn n=l (5.1) be in the class n. Then (5.2) and (5.3)

Lemma 3 [15]. Let the function w(z) defined by (5.1) be in the class n. Then lc2 - vc?I :'.S max {1, lvl} (5.4) for any complex number v. Equality in (5.4) may be attained with the functions w(z) = z2 and w(z) = z for lvl < 1 and lvl ~ 1, respectively . .

Theorem 4. If a function f(z) defined by (1.1) is in the class s>-(a:, ,B, A, B), ,B :f= ( A~B), then (a) for any real numberµ, we have (B-A).B(~-a)cos>.. [cos>. {(B - A),B(l - a)(l - 2µ) + [(B - A),B + AJ} + l[(B - A),B + A] sin >.I], if < -l+A+(B-A},B(2-a) µ - 2(B-A),B(l-a) ' (B-A).B(~-a)cos >.. [cos),+ l[(B _ A),B + A) sin>.!], f -l+A+(B-A),B(2-a) < < l+A+(B-A),B(2-a) i 2(B-A),B(l-o) - µ - 2(B-A),B(l-a) '

Theorem 5. If a function f(z) defined by (1.1) is in the class c>-(a, ,B, A, B), ,Bf. ( A~B ), then (a) for any real numberµ, we have (B-A)/1(~-a)cos>.. [cos>. {(B -A),B(l - a)(l - ~µ) + ((B - A),B +A]}+ l[(B - A),B + AJ sin .,\I], J < 2[-l+A+(B-A)/1(2-a)] i µ - 3(B-A)/1(1-a) ' (B-A)/1(~-a)cos>.. (cos>.+ l((B - A},B + A] sin .,\I], J 2[-I+A+(B-A)/1(2-a)] < < 2[l+A+(B-A)/3{2-a)] i 3(B-A)/3(1-a) - µ - 3(8-A)/1(1-a)

Theorem 6. Let the function f(z) defined by (1.1} be in the class s>..(a, ,B, A, B). (a) If (k - 1) ,B(l - a)(k - a) cos 2 >. >, n .. ,., n { (k - 1)(1 - A2) - (B - A),B[(B - A),B + 2AJ · ·[(1 - a) cos2 >. + k - 1]}, (5.21)

Corollary 5. Let the function f(z) defined by (1.1) be in the class c>.(a, (3, A, B). Then, under the hypotheses (5.21) and (5.22). 1 n lanl :s; 1 II I [(B - A)/3 + AJ[(k - 2) + (B - A)/3(1 - o:)e-i,\ cos >..I , (5.38) n. k=2 for n = 2, 3, ... , N + 2; and l N+3 I an I :s; 1 ,.. r , 1 \ ._ , _ n II I[ ( B - A) /3 + A][ ( k - 2) k=2

Theorem 7. ,-s.r.5>..(a,,B,A,B), is the smallest positive root r0 of the equa- tion [(B - A),B + A] {(B - A),B(l - a) cos(, - >.) cos>. - [(B - A),B + A] cos 1} r2 - (B - A),B(l - a)r cos >.+cos,= 0. (6.5) The result is sharp.

Corollary 6. 1-c.r. c>.(o:, /3, A, B) is the smallest positive root To of the equa- tion (6.5). The result is sharp for the function f(z) given by { {1 - [(B - A)/3 + A]z} -<B-1/:~.:;;;J~:;;,.. co•;,.. , J'(z) = . exp [-A(l - o: )z e-i>. cos>..] , ~ being defined ( as before) by (6.12). /3 =/= ( A~B), /J=(A~B), (6.13) 7. Radius of Starlikeness and Convexity We first state and prove

Theorem 8. The sharp radius of starlikes of the class s>.(a,/3,A,B), /3 =I= ( A~B ), is given by Ts =2 { (B - A)/3(1 - a) cos>..+ [(B - A)2 /32(1 - o:)2 cos2 .A - 4[(B - A)/3 + A]2 . 1 -1 . [(B-A)/3(1-o:)cos2 >.. _ ]]2} [(B - A)/3 + A] l (7.1) The expression in (7.1) is real and finite only when /3 =I= ( A~B) and such that (B - A)2 /32(1 - a)2 cos2 >.. > 4[(B - A)/3 + AJ2 [(B - A)/3(l - o:) cos 2 ).. - 1] (7 2)

Corollary 7. The sharp radius of convexity of the class c>..(a, ,B, A, B), ,B # (A~8), is given by (7.1). The result is sharp for the function J(z) given by (5.41), ~ being defind (as before) by (7.3). References [1] 0. P. Ahuja, "Certain generalization of the Robertson functions," Yokohama Math. J. 31 (1983), 5-11. [2] M. K. Aouf, "Bounded p-valent Robertson functions of order oi," Indian J. Pure Appl. Math. 16 (1985), no. 7, 775- 790. [3] M. K. Aouf, "Bounded spiral-like functions with fixed s

Definition 1. A function f(z) E Sis in the class SA(a, /3, A, B) if and only if the inequality ~1 7w - 1 < 1 (1 4) (B - A)/3( zJ~H) - 1 + (1 - a)e-iA cos .X) + A( zJ~~)) - 1) · holds for some .X, a, /3, A and B (I.XI < f; 0 ~ a < 1; 0 < f3 ~ 1; -1 ~ A < B ~ 1; 0 < B ~ 1), and for all z E U.

Definition 2. A function f(z) ES is in the class CA(a,/3, A, B) if and only if the inequality =-Ll!l t'(z) I < 1 (B - A)/3( zf:~~)) + (1 - a)e-iA COS .X) + A zJ:(~)) holds for some .X, a, /3, A and B (I.XI < i; 0 ~ a < 1; 0 < /3 ~ 1; -1 ~ A < B ~ 1; 0 < B ~ 1), and for all z E U. · (1.5)

Definition 3. If J(z) E 51 and hi < i, then the ,-spiral radius of J(z) is (6.1) Definition 4. If F C 51 and bl < i, then the ,-spiral radius of F is , - s.r.F = inf [, - s.r.{f(z)}]. /EF Also in [22], .. Mogra introduced the concept of ",-convex radius" as follows: (6.2) Definition 5._ If J(z) E 5, the class of analytic functions in U, and 1,1 < i, then the ,-convex radius of J(z) is 1 - c.r.{f(z)} = sup { r: Re { ei-r(l + z;:~~~))} > 0, lzl < r}.

Definition 6. If G C 5 and 1,1 < f, then the ,-convex radius of G is , - c.r.G = inf [, - c.r.{f(z)}]. /EG (6.4)