Inequalities for the Associated Legendre Functions

In this paper bounds for the associated Legendre functions of the first kindPmn(x) for realx∈[−1,1] and integersm,nare proved. A relation is derived that allows us to generalize known bounds of the Legendre polynomialsPn(x)≡P0n(x) for the Legendre functionsPmn(x) of non-zero orderm. Furthermore, upper and lower bounds of the typeA(α,n,m)⩽maxx∈[−1,1]|(1−x2)α/2Pmn(x)|⩽B(α,,n,m) are proved for all 0⩽α⩽1/2 and 1⩽|m|⩽n. Forα=0 andα=1/2 these upper bounds are improvements and simplifications of known results.