A NOTE ON NEIGHBORHOODS OF ANALYTIC FUNCTIONS HAVING POSITIVE REAL PART

Let P denote the set of all functions analytic in the unit disk D {z lllzl < 1} having the form p(z) + s pkzk with Re{p(z)} > O. For a > O, let k--1 Na(p) be those functions q(z) i + S qkzk analytic in D with s. Ipk -qk I.< 6. We k=l k=l denote by P’ the class of functions analytic in D having the fore p(z) 1 + pkzk k=l with R [zp(z)]’} > O. We show that P’ is a subclass of P and detemine a so that N(p) = P for p P’.


I. INTRODUCTION
Let denote the class of functions f analytic in the unit disk D {z Ilzl < 1} with f(O) 0 and f'(O) i.For f(z) z + T. ak zk in Z and a > O, let the k=2 a-neighborhood of f be given by Na(f) {g(z) z + s. bkzk _Z kla k bkl,< 6}.k=2 k-2 For h(z) z, Goodman [I] has shown ihat Nl(h) = S* where S* denotes the class of univalent functions in /z which are starlike with respect to the origin.St. Ruscheweyh if f(z) z + E akzk lies in C, where C denotes the class of proved that convex k=n+l univalent functions in , then Na(f)= S* for a n 2-2/n.Fournier [3] found that if C were replaced by =(g cllZg:'Iz)l < then N a (f)= T for a n e-1/n.Brown [4] extended the results of St. Ruscheweyh and n Fournier and provided simpler proofs.We shall focus on a class of functions directly related to S* and to other classes of univalent functions.Let P denote the class of p(z) 1 + .pkz k with Re{p(z)} > 0 for functions analytic in 1 having the form k=1 zl < I.This family is usually called the Carathodory class.For f in Y, recall that f S* if and only if p(z) zf'(z)/f(z) lies in P.
Let P' denote the class of functions analytic in zl < 1 having the form p(z) + Z pk zk with Re{[zp(z)]'} > 0 for Izl < I.In this paper we shall define k=1 a neighborhood of p P' and determine ; > 0 so that N(p)= p. Hence, by (2.1) p P and P'c P. Now let us establish a criterion for a given function to belong to P. By (2.1) 1+z l+z q c P if and only if q(z)-.Since is univalent, then q c P if and only if iO q(z) + e iO for O< O< 2 and Izl < I.That is, for O< B< 2, Izl < 1.
We can express (2.3) in terms of convolutions.Let f and g be analytic in the k unit disk D. Recall that if f(z) ; a.z k and g(z) ; b,z then the convolution k=O K k=O K (or Hadamard product) of f and g, denoted by f-g, is f,g akbkzk.

+ e l+w
We define a 6-neighborhood of p for p P.
For any p(z) 1 + z pk zk in P and 6 >, O, the S-neighborhood of p, DEFINITION.k=1 denoted by N(p), is N(p) q(z) 1 k:lqk z kE=llPk qk .<Our main result is the following theorem.p(z) 1 + E pk zk belongs to P', then N6(p)=P, where  PROOF.Let p P'. Then by Lemma i, z(p,h 8) is univalent.For fixed 0 < r < I, choose Zo with IZo r such that mini z( *h 8 Izi )1 IZo(P*h )(zo)l.
Since z(p*hs)is univalent, the preimage L of the line segment from 0 to Zo[(P*hs)(Zo)]L.The function g(r) -In (1 + r) I is decreasing for r > 0 if g'(r) -2 In (I + r)+ r'('l'2 r} < O.It is not difficult to show that r (I + r) In (i + r) ,< 0 for r > O, from which it follows that g'(r)< 0 for r > O. Hence p*hol > 2 In 2 i.This completes the proof of Lemma 3. Now we may prove the theorem.THEOREM).Let p(z) i + pk zk E P' and let a be as in Lema 3. e PROOF (OF k=1 want to show that every q N(p) belongs to P, where q(z) 1 + qk is an arbitrary k=l but fixed function in Na(p).Hence, Ip k qkl ,< a. Observe that k=l h O*ql (ho*P) + ho* (q P)I > Ihg*pl Ihg*(q P)I ig -e k k=l 2 (qk Pk )z > az q k pk > O.

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 If p c P', then z(p*h O) is univalent for each 0 < 0 < 2. PROOF.Fix O< O< 2. Then [z(p,ho)]' zp(z)), i + e iO] ei@ -iO =- is an arc inside ]zl # r.Hence, for lz] r we have Iz(p*ho)l > IZo(P*ho)l [z(p*%)] Idzl > fO r [z(p*hla)] 'lldzl- Accordingly, we apply Lemma 2 to get [P*hla] (z)l > E [z(p*hla)]

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation