ON SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF HIGHER

The classes Tk(O), 0 0 2, of analytic functions, using the class Vk(O) of functions of bounded boundary rotation, are defined and it is shown that the functions In these classes are close-to- convex of higher order. Covering theorem, arc-length result and some radii problems are solved. We also discuss some properties of the class Vk(P) including distortion and coefficient results.

We now prove.
The class Pk(p) is a convex set. PROOF.
Let Vk(0) denote the class of analytic and locally univalent functions f in E with normalization f(0) -0, f'(0) and satisfying the condition When o-0, we obtain the class V k of functions with bounded boundary rota- tion.The class Vk(p) also generalizes the class C(p) of convex functions of order .
It can easily be seen [l] that f e Vk(o) if and only if there In the followlng, we will study the distortion theorems for the class Vk(o).We where Re a>O and Re(c-a)>O.These functions are analytic for z.E [4].
In addition, we define the functions and Hi, M 2 are as defined in (2.2).
This result is sharp.

PROOF.
Using (2.1) and the well-known bounds for IF'(z)l with FCVk, see Let d r denote the radius of the largest schlicht disk centered at the origin contained in the image of Izl < r under f(z).Then there is a point  where a,b,c and M 2 are respectively defined by (2.4) and (2.2).
Similarly we can calculate the lower bound for If(z) and this establishes our result.

2
We note that fa is univalent for a < (l-o)(k-2) since Vm consists of univalent functions for 2 m 4. Hence f is unlvalent even if f is 2 not unlvalent provided a < (l-o)(k-2)" Using tile standard technique, we can easily prove the following.

K.I. NOOR
The proof is straightforward and follows immediately from the defini- tion and Theorem 1.5.
Furthermore it can easily be shown that if f c Vk(O) then f is con- vex of order 0 for Izl ( r where r is given by (2.8).

VIlE CLASS
A class T k of analytic functions related with the class V k has been introduced and studied in [5].We now define the following.
be analytic in E. Then fcTk(o), k ) 2, 0 p<l, if there exists a function gcVk(p) such that f'(z) g'(z) e P for z e E.
Note that Tk(O) T k and T2(O) is the class of close-to-convex functions.
This result is sharp.
It Is well-known that for h e P (3.1)Thus, using (3.1) and (2.Let g e Vk(o) and let g'(z) c Pa, l" Then f is a convex function of order p for [z < r where r c (0,I) is the least positive root of the equation We can write ,( Thus r e (0,I).
When o=O, g V k.Then f is convex for Izl < r =F.For k-4, V k consists of unlvalent functions and in this case r =-This result is proved in [8].
For a O, k 4 and p 0, we obtain the known result r 3 22 of Ratti [9] and when k 2, we have the well-known result giving us the radius of convexity for close-to-convex functions.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning used an easily establtsled .qharpresult Icn lk(1-o), for all nl.
l-o)r + (l-2o)r + k(l-p)r + (l-2o) (l_u)C-a-I (l_zu)-b du, r(a)r(c-a) 0 I'-o] , such that If('-)1 dr-The ray from 0 to f(z lies o o o entirely in the image of E and the inverse image of this ray is a curve in Thus d If(z )1 ; c-1) -r G(a b; c-rl) 0 a M2(a,b c,r), z f (z) f (f,())c dd a 0 (2.7) for fV k (0) This problem has been studied for the class of univalent normalized functions in E and for the close-to-convex functions, see [3]./e have 284 K.I.NOOR TIIEOREH 2.2.Let .V k(o), 0 0 < l, k 2 and let a, O<a<l be given.Then f V for mla(l-p)(k-2)+2].

THEOREM 2 . 5 .
Let f c Vk(p), o $ 1/2.Then f maps [z] < r 0 onto a convex domain where r 0 is given by (I.6).The function f defined by (2.6) same way as in Theorem 2.1, we obtain the required result.REMARK 3.1.When 0=0, fcT k and since in this case b 0<1, c l+a-b, we have G(a,b; c, -1) 1. Letting r 1, with 0-O, in Theorem 3.1, we see that the image of E under functions f in T k constalns the schllcht disk Iz[ < k+'-" We now give a necessary condition for a function f to belong to the class Tk(O).10 THEOREM 3.2.Let f c Tk().Then, with z re and 01< 02; O(o<l, K.I.NOOR p(z) + e l. + 2 which satisfy the InequalityThe class P has been introduced in[8] and it is shown there that, for p , gl V k, P Pa, So fi(z) -O] )(l-O) Re gi(z) 1p(-)- Using Theorem 1.4 with p 0 and (3.4), we have the required result.

Furthermore
2, we have the class P of functions with positive re'a1 part.The

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