ON CLOSE-TO-CONVEX FUNCTIONS OF COMPLEX ORDER

The class S*(b) of starlike functions of complex order b was introduced and studied by M.K. Aouf and M.A. Nasr. The authors using the Ruscheweyh derivatives introduce the class K(b) of functions close-to-convex of complex order b, b 0 and its generalization, the classes Kn(b) where n is a nonnegatlve integer. Here S*(b) c K(b) Ko(b). Sharp coefficient bounds are determined for Kn(b) as well as several sufficient conditions for functions to belong to Kn(b). The authors also obtain some distortion and covering theorems for Kn(b) and determine the radius of the largest disk in which every f K (b) belongs to K (I). All results are sharp. n n


I. INTRODUCTION.
Let A denote the class of functions f(z) analytic in the unit disk E {z: zl < I} having the power series f(z) z + [. amZ z e E.
(1.1) m=2 Aouf and Nasr [1] introduced the class S*(b) of starlike functions of order b, where b is a nonzero complex number, as follows: S (b) f: f e A and Re + f(z) > 0, z e E}.
We define the class K(b) of close-to-convex functions of complex order b as follows: f K(b) if and only if f e A and zf' (z)   Re {I + g; I)}> 0, z E, (1.2) for some starlike function g.
The classes Rn, n N O and where N O is the set of nonnegatlve integers, were introduced by Singh and Singh [2], f R if and only if f e A and .an g(z) .b z n then f(z)* g(z) a b z 0 0 n 0 nn The operator D n is referred to in Ai-Amiri [3] as the Ruscheweyh derivative of order n.
Note that R 0 is the familiar class of starlike functions, S*.More, it is known [2] that Rn+ 1CRn, n NO, and consequently R n consists of functions starlike in E.
Let Kn(b) n E NO, b is a nonzero complex number, denote the class of functions In particular one can look at the work of Ruscheweyh [4].
Section 2 determines coefficient estimates of functions in Kn(b, n e N O In section 3, we obtain some distortion and covering theorems for Kn(b) and several sufficient conditions for functions to be in Kn(b).The radius of close-to-convexlty for the class of close-to-convex of complex order b is also determined in section 3.

COEFFICIENT ESTIMATES.
In this section, sharp estimates for the coefficients of functions in Kn(b) are determined in Theorem 2.1.First, we need the following lemmas.
LEMMA  Since g E Rn, D n g(z) E S Thus, using the well known coefficient estimates for starlike functions one gets and the proof is complete.
for some g e R and where w e A such that w(0) 0, w(z) and Iw(z)l Using Clunle's method, that is to examine the bracketed quantity of the left-hand side in (2.6) and keep only those terms that involve z k for k m-for some fixed m, moving the other terms to the right side, one obtains Let z re 0 < r < 1. Computing f (z) (z) dz for both expressions of 0 in (2.7) and using Iw(z)l < we get .In particular, when m 2 we have The proof of the lemma is complete.This result is sharp.An etremal function is given by (2.3).We use the second principle of finite induction on m to prove (2.9).n! 2(b) is true as shown in (2.8) Now For m 2, 12a 2 -c21 ( <n + I), 21bl (n + I) assume (2.9) is true for all m (p.Taking m p + in (2.4), we get I(P + l)ap+ Cp+I12 Now using (2.9) since k p, the above yields n! p! !l(p+l)ap+ Cp+ll 2 , 4 (n + p) 1512 Applying the principle of mathematical induction on p, it is easily seen that the sum of the last two terms appearing in the bracketed expression in the right hand slde of the above is equal to !2 a is a close-to-convex function of complex mffi 2 m order b, then I%1 <-)lbl + I.This result Is sharp.
REMARK 2.2.For b 1, Corollary 2.1 is reduced to the well known coefficient bounds for the close-to-convex functions due to Reade [5].
Next we have two theorems that provide sufficient conditions for a function to be in Kn(b).THEOREM 2.2.

Let f e A and n E N O
If any of the following conditions is PROOF.The proofs follow by choosing g as below: For n e N0' each of the following m-2 m conditions is sufficient for f to be in Kn(b). (i) am+1,l lbl, where a I, We prove the sufficiency of part (1) since the proofs of the remaining parts are slmilar to the proof of (1).
Thus (2.13) is established and the proof of the sufficiency of part (i) is complete.REMARK 2.3.For n 0 and b I, Theorems 2.2 and 2.3 are reduced to theorems of Ozakl [6].
The objective of this section is to obtain some distortion theorems for the class The radius of the largest disk E(r) {z/Is < r}, 0 < r 4 such that if Kn(b).
f e K (b) then f K (I) can be determined as a consequence of one of those results..,. f.z..' <-3 (1 + r) (1 r) This result is sharp.An extremal function f is given by (2.1).PROOF.Let f K (b).Then (I.5) implies for some g e R The definition of R n implies D n g(z) is a starlike function.Hence by the well known bounds on functions which are starlike in E, we get for zl r < r ID n g(z)l < r (I + r) 2 (I r) 2 "(1 r) 3   For the proof of Theorem 3.2, we need the following well known result [7; p. 84]   concerning the class P of functions p(z) which are regular in E such that p(0) and Re p(z) > 0, z e E. where p e P. Hence (3.5) can be obtained by substituting p(z) in (3.4).
It is interesting to note that the result in Theorem 3.2 does not depend on the value of n.
Also, it can be used to solve the problem concerning the radii of Kn(b) in Kn(1).It can be shown that this disk lles in the 2 right half plane if r < r'.This completes the proof of Theorem 3.3.REMARK 3.1.Taking n 0 in Theorem 3.3, one can see that, r' is the sharp radius of close-to-convexlty for close-to-convex functions of complex order b.
stands for the Hadamard product of power series, i.e.

(3. 3 )Using ( 3 . 2 )
together with(3.3)one can get (3.1) and the proof of the Theorem 3one can immediately obtain the followlng corollarles, respectlvely.COROLLARY 3. I.If f is a close-to-convex function of complex order b where COROLLARY 3.2.If f is a close-to-convex function then for zl r < I,

LEMMA 3 . 1 . 2 , 2 .
Let p e P. Then for Izl r < I, Let f e K (b), n N O Then for some g e R and for Izl r < I, result is sharp.An extremal function is given in (2.1).PROOF.fK (b) implies that for some g e R

THEOREM 3 . 3 .
Let n e N O If f e K (b), then f e K (I) for Izl < r' where n n r This result is also sharp.An extremal function is given in (2.1).PROOF.Let f K (b).Then according to Theorem 3.2 there is some g e R