BOUNDED SPIRAL-LIKE FUNCTIONS WITH FIXED SECOND COEFFICIENT

Let F (c,B,M) (o ( p ( I, I(,I I/2 ), denote the P class of functions f(z) which are regular in U- {z:Iz < I} and of the form

" < "" (l-B)eo s In this paper we have found the sharp radius of Y-splralness of the functions belonging to the class F (a,8,M).P KEY WORDS AND PHRASES.Spfrallike, bounded functions, radius of Y-splralness.
1980 AMS CLASSIFICATION CODES.30A32, 30A36.I. INTRODUCTION.Let A denote the class of functions which are regular and univalent in the unit disc U {z:[z < I} and satisfy the conditions f(O) 0 f'(O)-l.

Let
F(e,8,M)(Ie <--, 0 < 8 < and M > I/@ denote the class of bounded e-splralllke functions of order 8, that is f e F(,8,M) if and only if for fixed M, la zf" (z) e 8cos -i sin f(z) M < M, z e U.
Thus without loss of generality we can replace the second coefficient a 2 of f(z) e F(=,,M) by la21 e Let F (a 8,M) denote the class of functions f(z) z + la21 e-laz 2 p which satisfy (I.1),where la21 p(t + o)(l 8) cos a.In view of (1.2) it follows that 0 ( p I.
It follows from (I.I) and (1.3) that g(z) O (a,B,M), if and only if zg'(z) e F (a,B,M).We note that by giving specific values to p,a,8 and M, we obtain the following important subclasses studied by various authors in earlier papers: (i) FI(a,,M) FM(a,) and GI(a,8,M) GM(a,8), are respectively the class of bounded splralllke functions of order and the class of bounded Robertson functions of order 6 investigated by Aouf [1] and FI(a,O,M) Fa, M and GI(a,O,M) G are respectively the class of bounded spirallike functions and the class of bounded Robertson functions investigated by Kulshrestha [2].
In this paper we determine the sharp radius of T-spiralness of the functions belonging to the class F (a,8,M), generalizing an earlier result due to Kulshrestha P [2], Llbera [4], Umarani [5,3].
he technique employed to obtain this result is similar to that used by McCarty [6] and Umaranl [3].
Since f(z) z + la21e-laz 2 + we obtain w(z)=pz + z(z), where #(z) is analytic in U, (0) Izl r under g(z) is a disc and h(z) is a billnear transformation, then zf'(z) zf'(z) f(z) is subordinate to (hog) (z).That is, the image of zl r under f(. is contained in the image of Izl r under (hog)(z).
This completes the proof of the lemma.REMARK I.
If YffiO in the above theorem, we obtain the radius of starllkeness of the class (----)(I-8) cos a -I.