CO-CONVEXIAL REFLECTOR CURVES WITH APPLICATIONS

. The concept of reflector curves for convex compact sets of reflecting type in the complex plane was introduced by the authors in a recent paper (to appear in J. Math. Anal. and Appln.) in their attempt to solve a problem related to StieltJes and Van Vleck polynomials. Though, in the said paper, certain convex compact sets (e.g. closed discs, closed line segments and the ones with polygonal boundaries) were shown to be of reflecting type, it was only conjectured that all convex compact sets are likewise. The present study not only proves this conjecture and establishes the corresponding results on Stleltjes and Van Vleck polynomials in its full generality as proposed earlier by the authors, but it also furnishes a more general family of curves sharing the properties of confocal ellipses.

The present study has been motivated by a recent conjecture (cf. authors [I, concluding Remarks (1)]) that every convex compact subset of the complex plane is of reflecting type. This arose while solving a problem related to stleltjes and Van Vleck polynomials. In this paper we are able to prove this conjecture by the introduction of a nice function v: R_+ (, and R_+ denote the set of all complex real and nonnegative real numbers, respectively). In fact, Section 2 is primarily intended to establish some relevant properties of the function v that is solely responsible for materializing, in Section 3, the family of the so-called co-convexlal reflector curves needed to prove the said conjecture. Besides, this family of reflector curves does present an interesting geometrical feature in as much as itprovldes an analogous theory of confocal ellipses under very general conditions. Finally, section 4 highlights certain applications of the theory of co-convexial reflector curves by obtaining some new results on the zeros of stleltjes and Van Vleck polynomials, some of which were only predicted in [I] and [2].
Before proceeding further, it is desirable to explain certain notations and terminology to be used later. Unless mentioned otherwise, K denotes a convex compact zK.
It is known (cf. [3,Thm. 12.20]) that K has a rectifiable boundary (with length denoted by IKI). The following special case of a theorem in Valentine   Observe that z z(K,T for all zK. However, the fact that every line L (not z cutting K) has a unique reflector point of K in L follows from the following lemma. LEMMA 2.4. Given .convex compact subset K o_f, le__t L be a directed lin___e (no__ cuttin K with a preassigned positive direction. For each zeL, i_f a(Az), a(A) and a(B z) denote the angles which Az, A'z ---and Bz, respectively, make with the positive direction of L, then (a) each of a(Az), a(A) and a(B z) increases strictly and continuously with range For proving our next lemma, we introduce the following notations: Given a,b K, we write (a,b) for the portion of K from a to b described in the clock-wise direction of K. The length of (a,b) will be denoted by LEMMA 2.  Next, we prove the following results. By a curve we mean a continuous arc whose initial point coincides with its terminal point. Therefore, every point of Cz is a boundary point of K(Cz). That is, Cz is a convex curve, and the lemma is established.
A regular Jordan curve C, lying outside a nonempty convex compact set K, is called a refiector curve for K (cf.[l, Definition 2.1]) if the normal at every point cC is along B the bisector of the angle a(c,K). A nonempty convex compact subset K of C is said to be of reflecting type (cf.[l, Definition 2.3]) if It has a unique convex reflector curve, enclosing K, through every point zK (it may be noted that any two such where (z) is a polynomial of degree at most (p-2) and where aj, aj are complex constants. It is known (cf. [6], [7,p.36]) that there exist at most C(n + p 2, p 2) polynomial solutions V(z) (called Van Vleck polynomials) such that, for #(z) V(z), the equation (4.1) has a polynomial solution S(z) of degree n (called stleltjes polynomials. ).
The differential equation (cf. [2], [8], where (z) is a polynomial of degree at most (nl+n2+...+np-2) and where aj,ajs,bjt are complex constants, can always be written in the form (4.1) by expressing each fraction (in the coefficient of dw/dz) into its partial fractions. In fact, (4.2) is surely of the form (4.1) if n.=l for all j. It may also be observed (as in case of (4.1)) that 3 there exists at most C(n + n + n 2 + + np 2, n + n 2 + + np 2) polynomials V(z) such that, for (z)=V(z), the differential equation (4.2) has a polynomial solution S(z) of degree n. That is, there do exist StieltJes polynomials S(z) and Van Vleck polynomials V(z) associated with the differential equation (4.2).
For convenience, we shall write q max{nl,n 2 n }.
(4.3) P Throughout this section, unless mentioned otherwise, K will denote a nonempty convex compact subset of C. We write R K(C z) for each z K and call it the reflector z region for K determined by z (cf. [l], [2])o Given K and (0 < N ), we recall (cf. [7,p.31][l], [2]) that the s>arrg_haped region S(K,) is given by S (K,) {ze a(z,K) Z @}. Given K, y e [0, w/2) and an integer q a I, we write (cf. [ [2]).