COVERING OF RADIAL SEGMENTS FOR DOMAINS BOUNDED BY k-CIRCLES

Some theorems on radial segments is studied for ring domains bounded by k-circles.


INTRODUCTION.
We designate the chordal distance between the points w and w 2 in the extended complex w- plane C by q(wl, w2) that is q(Wl, W2) Wl w2 [/J(1 + [w112)(1 + w2 [ 2)   if w and w 2 are both finite, and q(wl, oo I I1 + I,12. We define the chordal cross ratio of quadruples Wl, w2, w3w4 in C by (1.1) (1.2) X(w 1, w 2, w 3, w4) q(wl' w2)q(w3, w4) q(wl w3)q(w2, w4 (1.3)A Jordan curve 7 in C is called a k-circle, where 0 < k < 1, if for all ordered quadruples of points on 7, X(Wl,W2,W3,W4) + X(w2,w3,w4,wl) _< l/kl (1.4)This definition of a k-circle was introduced by Blevins [1].It is well-known that a k-circle is a quasicircle (see [2]).One of the simplest k-circle is {w: arg w arcsin k}.Throughout the note, the value of arcsin is restected between 0 and r/2.We consider a class of C(k)of conformal mappings w=f(z) on an annulus A(R) {1 <[z [< R} whose images D(7)= f(A(R)) are ring domains 7 with inner boundary f(Iz 1): {]w 1} and outer ones k-circles % Let w and w 2 be the points on such that w rl ei and w 2 r2e i(O + ) with 0 _< 0 < r.In this note we will consider the problem when the minimum of the values rlr2 and 1/r + 1/r 2 are attained.Corresponding to our problem, recently analogous ones were discussed for the classes of conformal functions by Aharanov and Kirwan [3], and Blevins [4].They considered the classes of functions conformal in the unit disk but we will do in an annulus, using a simple and elementary method, while their methods are rather complicated.
In order to solve our problem the technique of circular symmetrization will be used. 2. LEMMAS.In this section we summarize the pertinent facts in the following lemmas 2.1-2.5 which are necessary to prove covering theorems in section 3.  (2.3) LEMMA 2.4 [4].Let D be a ring domain with inner boundary {Iw 1} and outer one 3' a k- circle.If 3' contains the point at infinity and a point w' with w'l a, then the circular symmetrization D* of D with respect to the positive real axis is contained in the domain D(k,a) {w: arg (w + a) < r-arcsin k} f3 {I w > 1}. (2.4) Now we prove the following lemma which plays important roles in section 3.
LEMMA 2.5.Let w f(z) be a function C(k) and 3" f(Iz I= R) contain the point at infinity.Then for the distance d(3',0) between the origin and % there holds the inequality d(3',0) _> a0, ( where a 0 is a positive constant uniquely determined from the relation Mod D(k, ao)= log R for fixed values R and k.The equality (2.5) holds if and only if D(3') is D(k, ao) except for a simple rotation around the origin.
PROOF.At first we will verify that the equation has a unique solution a a O. Mod D(k,a) is a strictly increasing function of a variable a.Since lima-.,ooModD(k,a)= cx and from Lemma 2.2 lim a ooMod D(k,a)= 0, there exist a and a 2 such that a < a 2 and Mod D(k, a 1) < log R < Mod D(k, a 2). (2.7) Continuity of the module holds for the sequence of ring domains bounded by a finite number of analytic curves from Lemma 2.3.Therefore the equation (2.6) has a unique solution a a 0 for fixed values k and R.
Let w' be a point on 3' such that w'l =d(3',0)(=a).We consider the circular symmetrization D*(3') of D(3') with respect to the positive real axis.Using Lemma 2.1, 2. and monotonicity of the module, we have a _> a 0 (2.12) which implies the desired inequality (2.5).Using Lemma 2.1 we conclude that equality in (2.5) holds if and only if D() is D(k, ao) except for a simple rotation around the origin.
In this section we will prove the following covering theorems on radial segments.
The equality holds if and only if f(A(R))is the image of D(k, ao) under the mapping PROOF.At first we estimate min 0 < 0 < r(rl + r2) and then we apply the result to proving the inequality (3.1).Without of generality we can assume (considering a rotation if necessary) min r(rl 0<0< and b, -bt 7. We consider a M6bius transformation C(w) (1 + fbw)/(w + fb) ( which maps f(A(R)) onto a ring domain D(7') whose boundary consists of inner boundary CI-1} nd outer one '.Since the chordal cross ratio is invaxiant under M6bius transformations, 7' is a k-circle.By the mapping (3.4), b and -bt axe transformed onto (1 + tb2)/(b + fb) and the point at infinity, respectively.
(3.8)By this transformation (3.8) D( 7) is mapped onto a domain D(7') whose boundary consists of inner boundary {ICI 1} and outer 7' a k-circle.Using Lemma 2.5 we have T. INOUE   which implies, substituting w rl eiO and w 2 r2 ei(O + r), ( + q)/(q + ) > a 0, and then rlr2 >_ a O (rl+r2)-l.Combining the relation (3.7) and (3.11) we have r lr2 > 2a0(a 0 + qao 2 1)-(a 0 + a02 )2. (3.9) is the image of D(k, ao) under the mapping (3.2), the equality in (3.1) holds for w l=a0+a-I and w 2= -a 0-a-l.On the other hand, considering the case when the equalities hold, we can easily conclude that the equality in (3.1) holds only if f(A(R)) is the image of D(k, ao) under the mapping (3.2) except a simple rotation around the origin.
As the application of Theorem 3.1 we have the following theorem.THEOREM 3.2.Under the assumption as described in Theorem 3.1, we have 1/r + 1/r 2 <_ 2/(a 0 + . 1). (3.13)The equality holds if and only if F(A(R)) is the image of D(k, ao) under the mapping (3.2) except a simple rotation around the origin.
ACKNOWLEDGEMENT.The author would like to thank Prof. T. Kubo for his helpful suggestion.

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
and equality holds if and only if D* is obtained from D by a simple rotation around the origin.

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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