International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 2, Pages 233-240
Co-convexial reflector curves with applications
Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Received 11 November 1985
Copyright © 1987 Abdallah M. Al-Rashed and Neyamat Zaheer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.