Inequalities for Legendre functions and Gegenbauer functions (Q807887)

The well known inequalities for the Legendre functions of the first kind \(P^{\mu}_{\nu}(x)\), \(\nu\),\(\mu\in N\), \(x\in [-1;1]\), are significantly improved. These inequalities have the disadvantage that the bounds of the Legendre functions tend to infinity as \(x\to 1\). Moreover, two new inequalities for the \(P_{\nu}^{-\mu}(x)\), \(\nu\),\(\mu\in R\), \(\mu >-1/2\), \(\nu\)-\(\mu\geq 2\) and the Gegenbauer functions \(C^{\lambda}_{\alpha}(x)\), \(\alpha\),\(\lambda\in R\), \(\lambda >0\), \(\alpha\geq 2\) are established. In these inequalities the bounds of the special functions are finite on the whole [-1;1].