GENERALIZED CLASSES OF STARLIKE AND CONVEX FUNCTIONS OF ORDER

We have introduced, in this paper, the generalized classes of starlike and convex functions of order by using the fractional calculus. We then proved some subordination theorems, argument theorems, and various results of modified Hadamard product for functions belonging to these classes. We have also established some properties about the generalized Libera operator defined on these classes of functions.


INTRODUCT ION.
Let A denote the class of functions of the form f(z) z + 7 a z n n n=2 (i.i) which are analytic in the unit disk U {z:Iz < I}.Furthermore, let S denote the subclass of A consisting of all univalent functions.A function f(z) of S is said to be starlike of order if Re(Zf'(z) f(z) > a (1.2) for some (0 = a < i) and for all z U We use S* () to denote the class of all starlike functions of order Similarly, a function f (z) belonging to S is said to be convex of order if we replace (1.2) by zf" (z) Re(l + > (1. 3) f' (z)   We use K() to denote the class of all convex functions of order Note that f(z) K() if and only if zf' (z) S*(d) and that S*() S*(0) S*, K() K(0) K and K() S*() (0 _<-< i).The class S*() and K() were introduced by Robertson [I], and studied subsequently by Schlld [2], MacGregor [3], Pinchuk [4], and others.
For our discussion, it is more convenient to use the following definitions which were employed recently by Owa [12 and by Srivastava and Owa [13].
DEFINITION 1.1.The fractional integral of order % is defined, for a functlon f (z) by z Is an analytlc function in a simply-connected region of the l-i Z-plane containing the origin, and the multiplicity of (z-) is removed by requiring log(Z-) to be real when z-> 0 DEFINITION 1.2.The fractional derivative of order l is defined, for a where 0 _<-I < 1 f(z) is an analytic function in a simply-connected region of the z-plane, and the multiplicity of (z-) is removed as in Definition i. 1 above.
DEFINITION 1.3.Under the hypotheses of Definition 1.2 the fractional derivative of order n+% is defined by Also let K(C,l) be the class of all functlons f(z) in S such that A(l,f) S*(,l) for < 1 and 0 <-< 1 We note that S*(,0) S*() and K(,0) K().Thus S*(,l) and K(,I) are the generalizations of the classes S*(e) and K() respectively.The classes S*(,I) and K(,I) were introduced by Owa [14].Recently, Owa and Shen [15] proved some coefficient inequalities for functions belonging to the classes S*(,I) and K(,I) Let T be the subclass of S conslsting of all functions of the form The classes T* (e,l) and C(,I) were studied by Owa [14 ], and the special cases T*(,0) and C(,0) were studied by Silverman [16].Thus the classes T*(,l) and C(,I) provide an interesting generalizatlon of the ones considered by Silverman [16].
In sections 2, 3 and 4, we shall prove several results for functions belonging to the generalized classes S*(,l) K(,l) T*(,I) and C(,I) We then introduce the class S*(,l;a,b) of functions in section 5.In the last section, we shall study a certain Integral operator defined on A.

SUBORDINATION THEOREMS.
Let f(z) and g(z) be analytic in the unit disk U Then we say that f(z) is subordinate to g(z), written f(z) ,< g(z), if there exists a function w(z) analytic in the unit disk U which satisfies w(0) 0, lw(z) < i, and f(z) g(w(z)).In particular, if the function g(z) is univalent In the unit disk U then f(z) .<g(z) if and only if f(0) g(0) and f(U) g(U) (cf. [17], [18]).
In order to prove our first theorem, we require the following lemma due to Miller, Mocanu, and Reade [19].
PROOF.Note that the function g (z) defined by maps the unit disk U onto the half domain such that Re (w) > This Implies from the definition of the class S*(,I) that Furthermore, the function g(z) is analytic wlth g' ( 0) 2 (l-s) M 0 and Taking l, y 1 in Lemma 2.1, we see that the function g(z) satls fies the hypotheses of Lemma 2.1.Thus we have z z which implies (2.5).
PROOF.Note that f(z) K(,I) f and only if A(%,f) S*(,l).
Consequently, on replacing f(z) by A(l,f) in Theorem 2.1, we have Theorem 2. In this section, we derive the argument theorems for functions belonging to the classes S*(,I) and K(e,l).
THEOREM 3. i.Let the function f (z) deflned by (i.i) be in the class S*(,l) (0 _<- In view of (2.6), we can write where w(z) is an analytic function in the unit disk U and satisfies w(O) O and lw(z) < 1 We note that the linear transformation maps the disk wl & z onto the disk.This completes the proof of Theorem 3.1.COROLLARY 3.1.Let the function f(z) defined by (i.i) be in the class S*(0,I) ( < i).Then arg f(z) where AO(l,f) zl/IDl+tf (Z) CO.'.OLLARY 3.2 Let the function f(z) defined by (i i) be in the class S*(e,l) (0 =< c < i; I < i).Then l-(l-2e) Izl IA(l,f) l+(l-2e)Izl l+Izl llf(z) l-lz where A(A,f) is given by (1.8).
PROOF.The proof is clear from (3.5).
Moreover it is easy to show that THEOREM 3.2.Let the function f(z) defined by (I.i) be in the class K(,A) (0 < i; A < i).Then We recall here the following two lemmas due to Owa [14] before state and prove our results of this section.
LEMMA 4.1.Let the function f(z) be defined by (1.9).Then f(z) is in the class T*(,I) (0 <= < I; A < i) if and only if for 0 <= < i and A < 1 The result (4.3) is sharp.
LEMMA 4.2.Let the function f(z) be defined by (1.9).Then f(z) is in With the aid of Lemma 4. i, we can prove THEOREM 4.1 Let the functions f. (z) (j 1,2) defined by (4.1) be in the class T*(Q,A) (0 _<- < i; A < i).Then the modified Hadamard product fl*f2(z) is in the class T*(Z(Q,A) ,l) where (2-e+l) The result is sharp.
PROOF.We use a technique due to Schild and Silverman [20].It is sufficient to prove that 7 for 8 <-8(Q,l).By using the Cauchy-Schwarz inequality, we know from (4.3) that ,n n= 2 Therefore we need find the largest 8 such that 7.

F(n+I) F(I-
In view of (4.7), we observe that it suffices to find the largest 8 such that we note that (4.10) gives where Since, for fxed a Finally, by taking the functions f. (z) (j 1,2) defined by we can see that the result is sharp.
By using the same technique and Lemma 4.

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The result is sharp.whlch shows that F (z) maps the unit disk U onto a domain whlch is contained in the right half-plane.Hence we complete the proof of Theorem 5.1.(5.9) 6. GENERALIZED LIBERA OPERATOR J (f).This operator J (f) when c is a natural number was studied by Bernardi [21]. c In particular, the operator Jl(f) was studied by Libera [22], Lvingston [23], and Mocanu, Reade and Ripeanu [24].It follows from (6.1) that In order to prove our theorem, we recall here the following theorem due to Jack [25].
LEMMA 6.1.Let w(z) be regular in the unit disk U with w(0) 0 Then, if lw(z)l attains its maximum value on the clrcle Izl r (0 _<-r < i) at a point z 0 we can write z0w'(z0) mw(z 0) where m is real and m->_ 1 Furthermore, we need the following lamina by Pascu [26].
LEMMA 6.2.If f(z) S*, then J (f) ( S*. c With the aid of Laminas 6.1 and 6.2, we prove THEOREM 6.1.Let the function f(z) defined by (i.i) be in the class S*(,I) S* (0 -< < I; i < i).Then the functional J (f) is in the class c s* (,l).
PROOF.Define the function w (z) by Then w(z) is a regular function in the unit disk U with w(0) 0.

F
the class C(,A) (0 -< < I; A < I) if and only if E F(n+I) F(1-A) {F(n+I) F(1-A)