An acknowledgment of priority

A. A. Bennett
1920 Bulletin of the American Mathematical Society  
+ i[«"(0 + 3gi(f)»(r)]«»(»)t^W. £(#, ÎJ) = 0 is therefore the necessary and sufficient condition that v(z) be factorable as the product of a solution of (5) by a solution of (6). Therefore, if S(v, v) = 0 the number of zeros of v(z) in a given closed interval (a, b) will be limited by Sturm's theorems concerning the zeros of solutions of (5) and (6). If, for example, the equations (5) and (6) have fa and k 2 for their respective indices of oscillation in (a, b) then every solution of (5) [(6)]
more » ... ution of (5) [(6)] will have hi or k\ + 1 [k 2 or k 2 + 1] zeros in the interval (a, 6), and every solution, v(z), of (3) for which S(v, v) = 0 will have fa + k 2 , fa + k 2 + 1, or &i + k 2 + 2 zeros in (a, fc). If, in particular, q 2 (z) < 0 and 0 < C < 9q 2 2 (z) throughout (a, 6) then the coefficients of u(z) and w(z) in (5) and (6) will be negative throughout (a, b) smdfa = k 2 = 0.* Therefore, #»(*) < OemdO < C < 9q 2 \z) throughout (a, 6) then no solution of (3) for which S(v, v) = 0 Aas more tfilem tfwo seros m (a, 6), and some such solutions have no zeros in (a, 6).
doi:10.1090/s0002-9904-1920-03304-5 fatcat:g7jttqngpbd4hblsanaituyfke