INEQUALITIES FOR THE ASSOCIATED LEGENDRE FUNCTIONS

In this paper bounds for the associated Legendre functions of the firs t kind P-n(m)(x) for real x is an element of [-1, 1] and integers m, n are proved. A relation is derived that allows us to generalize known bounds of the Legendre polynomials P-n(x) = P-n(0)(x) for the Legendre functions P-n(m)(I) of non-zero order pn. Furthermore, upper and lowe r bounds of the type A(alpha, n, m) less than or equal to max(x) (is a n element of) ([-1,) (1])\(1 - x(2))(alpha/2) P-n(m)(x)\ less than or equal to B(alpha, n, m) are proved for all 0 less than or equal to alp ha less than or equal to 1/2, and 1 less than or equal to \m\ less tha n or equal to n. For alpha = 0 and alpha = 1/2 these upper bounds are improvements and simplifications of known results. (C) 1998 Academic P ress.