FUNCTIONS STARLIKE WITH RESPECT TO OTHER POINTS

In [7], Sakaguchi introduce the class of functions starlike with respect to symmetric points. We extend this class. For 0 .< 8 i, let S*(8) be the s class of normalised analytic functions f defined in the open unit disc D such that Re zf’(z)/[f(z)-f(-z)) > 8, for some z e D. In this paper, we introduce 2 other similar classes S*(8), S (8) as well as give sharp results for the real c sc part of some function for f e S(8) S(8) and S (8) The behaviour of certain s sc integral operators are also considered.


8.
In [7], Sakaguchi introduced the class S* of analytic functions f, normalised s by (1.1) which are starlike with respect to symmetrical points.We begin by defining the class S* which is contained in K, the class of close-to-convex functions.
s' We now extend this definition as follows: S.A. HALIM DEFINITION 2.
A function f with normalisations .]J is said to be starlike of order 8, with respect to symmetric points if, and only if, for z e D and 0 .< 8< l, We denote this class by S* (8) and note that S* S*(0).s s s In the same manner, we define the following new classes of close-to-convex functions, which are generalisations of the classes in E1-Ashwah and Thomas [2].DEFINITION  A function f normalised by (1.1) is said to be starlike of order 8, with respect to symmetric conjugate points if, and only if, for z e D and 0 .< 8l, Re / .?_'.i._ .

RESULTS. THEOREM i. i8
Let f e S*(8), then for z re e. D, The result is sharp for fo given by f0(z) fo(-Z) 2z(l+z2) 8-1 To prove Theorem i., we first require the following lemma.
The Lemma now follows at once.

PROOF OF THEORY4 i.
Since f e S*(8), it follows that we may write s g(z) (z)-f(-z) 2 for g an odd starlike function of order 8.An application of Lemma 1 proves the Theorem.
Results analogous to Theorem i THEOREM 2.  The result now follows immediately.
Similarly, we have the following result, which we state without proof.THEOREM 3.

> z l+r
The resul is sharp for f(z) f(-) 2z(l+z) We now consider the results of some integral operators.In  Let M and N be analytic in D with M(O) N(O) 0 and let 6 be any real number.If N(z) maps D onto a (possibly many sheeted) region which is starlike with respect to the origin, then for z e D, ' Re N--(z > 6 ----->Re N--> 6, and PROOF OF THEOREM h. (2.2) ives, i: + + N' () Thus N(z) is starlike if, and only if a > -8.Furthermore, since iI Re N' (z) Re (z)-f(-) > " Lemma 2 shows that H e S* (8).
On using Lemma 2 it follows that H e S*(8) c THEOREM 6.
Let f e S* (8).Then H defined by sc H(z) a+--!l [z ta_l[f{t f(-:)]dt, For f e S* (8), (2.4) gives one can show that N e S* and hence using Lemma 2 the result follows.
DEFINITION 1.A function f e S if, and only if, for z e D, s Re f(Z)2f(iz) > O.
LEMMA i. ieLet g e S*(8) and be odd.Then for z re D a S*s(S).
S*(8), it is easy to see that, if C F(z) f(z)+f() then F a S*(8).from (2.1) that Using the same techniques as in the proof of Lemma i, it follows S.A. HALIM Re z [i] Das and Singh, obtained analogous results of the Libera integral operator.They proved that for f e S*(0), the function h given by s t-l[ f( t)-f(-t) ]dt h(zJ 70 also belongs to S*(0).s The result below generalises that of Das and Singh.
to S*(6) .forz D and a + 6 O. s We first require the following Lemma due to Miller and Mocanu[5].LEMMA2.