ON THE BOUNDS OF MULTIVALENTLY STARLIKENESS AND CONVEXITY

The object of the present paper is to prove some interesting results for the bounds of starlikeness and convexity of certain multivalent functions.


I, INTRODUCTION,
Let A(p) denote the class of functions of the form f(z) z p + anzn n=p+l (p E N {1,2,3,...}) which are arlytic in the unit disk U {z: Izl < i}.A function f (z) in the class A(1) is said to be a member of the class p if and only if it satisfies Re{f'(z)} > 0 (z e U).
It is well known that if f(z) belongs to the class P, then f(z) is univa- lent in U (cf. [13, [23).
Many results for this class were obtained, but the radius of starlikeness for the class p is not known.
Lewandowski [3] has proved that if f(z) belongs to the class p, then f(z) is starlike in Izl < 4 5 % 0.6568.This result has been improved in ([43, [53) as follows- If f(z) belongs to the class p, then f(z) is univalently starlike in zl < p, where p is the smallest positive root of the equation  f(z) Then we shall call a function f(z) p-valently starlike in Izl < r.
We denote by S(P) the subclass of (p) consisting of functions which are p-valently starlike in .
DEFINITION 2, Let f(z) e A(p) and Then we shall call a function f(z) p-valently convex in zl < r.Also we denote by C(p) the subclass of A(P) consisting of all p-valently convex functions in the unit disk U.
LEMMA I, (Ruscheweyh [6]) Let f(z) be in the class p, and assume that f'(z) is typically real in U. Then f(z) is univalently starlike in the unit disk U.

3, BOUNDS OF STARL!KENESS AND CONVEX!TY,
We begin with the statement and the proof of the following result.
Then f(z) is p-valently convex in and p-valently starlike in .
PROOF.Let F(z) f(p-l)(z)/pl.Then it is clear that F(0) 0 and F'(0) i.Also, since F(z) , Re{F'(z)} > 0 (z e ), and F'(z) is typically real in .A n application of Lemma i to the function F(z) The above inequalities imply that f(z) C(p) and f(z) S(P), respectively.
Thus we complete the proof of Theorem i.
Next, we prove THEOREM 2, Let the function f(z) belong to the class A(P), and (f(P-l) (z) /p l) e p. Then f(z) is p-valently convex in Izl < r(p), where p2+l 1 r(p) P PR00, Defining the function F(z) as in the proof of Theorem i, that is, F(z) (f(P-l)(z)/p), we have F(0) 0, F'(0) i and Re{F'(z)} > 0 (z ).Then it is well known that zF"(z) zf (p+I) (z) Izl '2 f(P) (z U)-Thus, it follows from the above that zf (p+I) (z) > 0 for zl < r(p).Making use of Lemma 3 leads to zf" (z) } for zl < r(p) which completes the proof of Theorem 2.
REMARK, We can not find out an extremal function of Theorem 2.
Applying the same method as in the proof of [5], and using Lemma 2, we have the following result.
THEOREM , Let the function f(z) belong to the class A(P), and Further, spending the same nner as in the proof of ([8], [9]), and using Lemma  Then f(z) is p-valently starlike in Izl < P2' where 02 is the smallest positive root of the equation log(9 4r 2 + 4r3 r 4) log9(l r 2) + Sin-lr and 0.933 < 2 < 0.934.REMARK.The above corollary is an improvement of the result in [i0, Theorem 6].
Finally, we derive TH0RM 5. Let the function f(z) belong to the class (p), and If (p+l)(z)l <__ klzlk-lpl where k is a positive real number.Then f(z) is p-valent in .
PROOF.From the assumption of Theorem 5, we see that for z g Uo This implies that Re{f(P)(z)} > 0 (z g ).By applying Ozaki's theorem [ii] to the function f(z), we conclude that f(z) is p-valent in .
2, we get the followimg theorem.HFORFM , Let the function f(z) belong to the class (p), and If (p)(z) p!l < P Letting pi in Theorem 4, we have COROLLARY.Let the function f(z) belong to the class A(1z) is univalently starlike in Izl < 2' where 2 is given as in Theorem 4. If(P)(z)pllIf(P+l)(t)dtlIzl