EFFECTIVENESS OF TRANSPOSED INVERSE SETS IN FABER REGIONS

The effectiveness properties, in Faber regions, of the transposed inverse of a given basic set of polynominals, are investigated in the present paper. A certain inevitable normalizing substitution, is first formulated, to be undergone by the given set to ensure the existence of the transposed inverse in the Faber region. The first main result of the present work (Theorem 2.1), on the one hand, provides a lower bound of the class of functions for which the normalized transposed inverse set is effective in the Faber region. On the other hand, the second main result (Theorem 5.2) asserts the fact that the normalized transposed inverse set of a simple set of polynomials, which is effective in a Faber region, should not necessarily be effective there.

. Moreover, the effectiveness properties in Faber regions, of the transpose of simple sets of polynomials effective in the same regions, have been investigated by Nasslf [2]. We propose therefore, as a complement of the investigation of Nasslf, to consider in the present paper (which is extracted from the PhD Thesis of Adepoju [3; Chapter II]), the effectiveness properties, in Faber regions, of the transposed inverse set of a given basic set of polynomials. In particular, we shall study here the extent of generalization, to Faber regions, of the result of Newns referred to above. It should be mentioned here that the reader is supposed to be acquainted with the theory of basic sets of polynomials, as given by Whittaker [4], as well as the properties of Faber regions and Faber polynomials as given by Newns [5; pp which is supposed to be conformal for tl > T, for some T > y. Suppose also that t (z) is the inverse transformation, which is conformal for Izl > T', say.
The following notations are adopted throughout the present work.
In our notation, a cap (^) over a set indicates that the set is the transposed inverse of the corresponding set. Our main concern in the present work is to establish a relationship between the effectiveness properties, in Faber regions, of a given set pk(z) and those of the transposed inverse set pk(z) which may not exist in D(C) for certain drms of C, (as for example when C is a Cassini oval).
In fact, we write G(y)' [max{ G(')}J, (2.2) and the normalized set Pk(Z) is derived from the given set p k(z) through the substitution Pk(Z) Pk(Z) (k > 0), where is any complex number satisfying < lI < oo.

(2.5)
It should be observed, beforehand, that the normalizing substitution (2.3) will lead to favourable result when C is a circle of centre origin. In fact, if the given set pk(z) is effective in D(R) R > 0, it can easily be shown that the transposed inverse set lk(Z) of (2.5) will be effective in D+(I[/R)

R2"
Suppose now that C is not a circle of centre origin, as we shall always assume in the subsequent work. Hence, we deduce from (1.2) and (1.3), that, for all numbers " for which T" < N" < the curve CE(. is well defined. With the above notation, and with the usual notation for functions classes (c. f.    Po (z) fo(Z) pk(z) =-y e + fk(z) (k >i).
It can be verified that this set is effective in D+(C) and that, for the transposed inverse set, we shall formally have Po (z) go(Z) pk(z)=-Okfo(Z) + fk(z) (k > i). Now, we can apply the formula (1.6) to deduce, in view of (1.5) that ZeCTl (3.12) where M is a positive finite number independent of k and T I is a number chosen so that o0" < T I < y and this is allowable for the foregoing form of the curve C. impossible at Zo D(C) and the set @k(Z) is not basic in D(C) as required.
4. PROOF OF THEOREM 2.1. The proof is similar to that of theorem i of [2], even more direct. In fact, let qk (z) be the set assoclatlated with the given set Pk (z) I' which is now given as the product set pk(z) lqk(z) I Ifk(z)    We thus conclude that the set k(Z) is effective in D(C) for the class of functions H(C.) and theorem 2. i is therefore established.

NONEFFECTIVENESS OF TRANSPOSED INVERSE SETS.
As supplement to the positive result involved in theorem 2.1 above, we shall establish in the present section a negative result which is in contrast with the favourable result stated in 2 concerning the case of disks with centre origin (p. 5, ii. 1-5).
Actually, we shall prove, in the second main result of the present work formulated in theorem 5. It should be first mentioned that the inequality (5.4) cannot be deduced from corollary 2.2 above since the set Ik(Z) is not simple.
PROOF. We first observe, from (2.1) and (5.3) that " (Iz) k+l; (k> O> Therefore, adopting the notation applied before, namely, Wo eim/ i l(Wo) (5.10) then, as in formula (6.8) of [2] we deduce from (2.6) that o1>7. (5.11) Furthermore if G(z) i/(w z) and if the basic series associated with the o Taylor expansion of G(z) is h=o k (z) then from (5.10) the analogue of (6.9) of [2] is wl fk(ne-i) ( when C is either a circle or a Cassini oval.
The proof of theorem 5.2 is therefore complete.

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 • 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