ON A GENERALIZATION OF CLOSE-TO-CONVEXITY

A class Tk of analytic functions in the unit disc is defined in which the concept of close-to-convexity is generalized. A necessary condition for a function f to belong to Tk, raduis of convexity problem and a coefficient result are solved in this paper.

Let V k be the class of functions of bounded boundary rotation and K be the class of close-to-convex functions.We generalize the concept of close-to-convexity in the following direction.n Definition.Let f with f(z) cz + Y. a z be analytic in E {z:Izl<l} Icl=l and n=2 n f'(z) 0. Then feTk, k>_2, if there exist a function geV k such that, for zeE f'(z) Re >0. g'(z) (1.I) It is clear that T 2 E K.
Using a method by Kaplan [2], we have fgT k.Then with z re i0 and 0 < @ THEOREM i.Let (1.2) REMARK i.From theorem i, we can interpret some geometric meaning for the class T k- For simplicity, let us suppose that the image domain is bounded by an analytic curve i0 C. At a point on C, the outward drawn normal has an angle arg[ei0f (e ].Then from k (1.2), it follows that the angle of the outward drawn normal turns back at most .This is a necessary condition for a function f to belong to T k.It will be inter- esting to see if this condition is also sufficient.RE,LARK 2. Goodman [3] defines the class K(B) of functions as follows.
n Let f with f(z) z + Y. a z be analytic in E and f'(z) 0. Then for B>0, feK(6), n=2 n if for z=re i6 and 61 < 62 We note that TkCK().
From remark 2 and results given in [3] for the class K(B), we have at once where F k is defined by, for zeE, and clearly F k e T k. (ii) where A (k) is defined by (2.1).This result is sharp for each n > 2.
i6 (iii) For z re 0 < r < i, We also need the following result.
Lemma 1 [4].Let ggV k.Then there are two starlike functions s I and s 2 such that for zgE g'(z) .gk4 geV k and Re h(z)>0.
Using lemma I, we know that there are two starlike functions s I and s 2 where k I and k 2 are two suitable selected close-to-convex functions.
It is well-known [i] that for starlike function seS, Then by a result of Golusin [6,p162], there exists a z I with Zll r such that for all z, z[ r, 2r 2 where Re h(z) > 0. Aso (2.8) by using Schwarz's inequality, lemma 2 and (2.7).
We now evaluate the radius of convexity for the class T k.
Then the radius R of the circle which f maps onto a convex domain is given by R 1/2k+2>-X/k)], The function F k defined by (2.1) shows that this result is best possible.In par= ticular, when k 2, R 2aV--,which is well known.This result also follows from the remark in [3,p.23 For geVk, it is well known [9] that, for z re 0<r<l, This gives the required result.
(i).We also note that the extremal function Fk(Z) defined by (2.1) is the same function as FB(z) defined by equation (2.6) in [3].As A. W. Goodman has pointed out that this function is sometime referred to as the generalized Koebe function.
(li).We conjecture that the class T k is a proper subclass of the class K(B) as defined in [3], since in the definition of T k, gV k and we know that gVk, 2<_k<__4, is convex in one direction and all the functions in one direction form a proper subclass of the class of close-to-convex functions.
(ii).It remains open whether T k is a linear in variant family.

THEOREM 2 .
Let feT k-(i) Denote by L(r,f) the length fo the image of the circle Izl r under f and by A(r,f) the area of f(Izl=r).Then for 0<_r<l, (a) L(r,f) <_ L(r,Fk) (b) A(r,f) _< A(r,Fk)