SOME ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF VIBRATING STRINGS

If a string with integrable density function p(x) is fixed at the points x = 0, x = a then the natural frequencies of vibration are determined by the eigenvalues of the Sturm-Liouville System These eigenvalues depend on the density function p(x) and we denote them accordingly by λn(p), In this work we investigate the nature of the density functions which yield the largest and smallest possible value for λn(p) assuming that the average value of the density p(x) defined by is restrictedin some manner. © 1969 by Pacific Journal of Mathematics.