A 2ND PROOF OF THE PAYNE-POLYA-WEINBERGER CONJECTURE

Without using product representations or elaborate comparisons of zeros we prove the two key properties of the Bessel function ratio J(p + 1)(j(p + 1, 1)x)/J(p)(j(p, 1)x) that we used to prove the Payne-Polya-Weinberger conjecture. In these new proofs we use only differential equations and the Rayleigh-Ritz method for estimating lowest eigenvalues. The new proofs admit generalization to other related problems where our previous proofs fail.