ON THE RADIUS OF UNIVALENCE OF CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS KHALIDA

We consider for a > 0, the convex combinations f(z) (l-a)F(z) + azF’(z), where F belongs to different subclasses of univalent functions and find the radius for which f is in the same class.


E {z:
zl < i} which are respectively univalent, close-to-convex, starlike, and convex.In [1,2], a new subclass C* of univalent functions was introduced and studied.
A function f, analytic in E, belongs to C* if and only if there exists a convex func- tion g such that for z g E, Re (zf'(z))' > O.
(i.i) g'(z) Tile functions in C* are called quasi-convex and C C* K S. It is shown [2] that f g C* if and only if zf' g K.Recently the functions called a-quasi-convex have been defined and their properties studied in [3].A function f, analytic in E, is said to be s-quasi-convex if and only if there exists a convex function g such that, for a real and positive f'(z) It has been shown [3] that F is s-quasl-convex if and only if f with f(z) (i-e)F(z) + zF'(z) is close-to-convex. (1.3) All s-quasl-convex functions are close-to-convex.
We shall now study the mapping properties of f: f(z) (i -)F(z) + zF'(z), e > 0, when F belongs to different subclasses of univalent functions.
THEOREM 2.1.Let F S* and > 0. The function is starlike in zl < ro, where This result is sharp. (2.2) PROOF.We can write (2.1) as and from this it follows that 1 where Re h(z) > 0, since F e S*.
From (2.4), we have Differentiating both sides of (2.5), we obtain Now, using the well-known result [4], lh'(z) -< {2Re h(z)}/(l r2), Izl r, we have From (2.1) and (2.3), we have Using (2.7), we have from (2.6) Re hi ){(_.z._. (2.8) The right hand side of (2.8) is positive for r < r where r is given by (2.2).This o o result is sharp as can be seen by (2.9) where Let f e C, then f, given by (2.1), is convex for zl < r where r o o is given by (2.2).The proof follows on the same lines as in Theorem 2.1.See also [5] and [6].
REMARK 2.2.In [6], Nikolaeva and Repnina treated the same problem, with a dif- ferent notation, for the convex and starlike functions of order 8. Theorem 2.1 follows from their result when we take '8 0 for 0 < < i.On the other hand, our proof of Theorem 2.1 is much simpler and the result holds for all > 0.
As an easy consequence of (1.3) and Theorem 2.2, we have the following.
COROLLARY 2.1.Let F e K and f(z) (i e)F(z) + azF'(z), a > O. Then F is e- quasi-convex in zl < r This means that the radius of a-quasi-convexity for close- O to-convex functions is given by (2.2).
PROOF.Since F C*, there exists a G e C such that for z e E, Re (zF H(z), then from (2.10), we have Since from Theorem 2.2, the function (l-)tt(z) + ztt'(z) It(z))' be--.
longs to K with respect to a convex function g: g(z) (l-e)G(z) + ezG'(z) in zl < r so f is in C* for zl < r where r is given by (2.2). 0 0 0 REMARK 2.4.For F e C* and 1/2, Theorem 2.3 has been proved in [1].
We now deal with a generalized form of (i.i) by taking g to be starlike and prove the following.