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Some bounding inequalities for the Jacobi and related functions. (English) Zbl 1131.26005

The paper describes establishment of bounding inequalities for the Jacobi function as a consequence of reasonably sharp inequalities for the classical Laguerre functions, given in Section 2 of the paper in the form of four lemmas. By virtue of hypergeometric representations of the classical Jacobi function \(P^{(\alpha,\beta)}_\upsilon(z)\) \((\upsilon\in C)\) of the first kind and the classical Laguerre function \(L^{(\mu)}_\upsilon(z)\) \((\upsilon\in C)\) in terms of \({_1F_1}(\cdot)\), the Eulerian integral is written and then appealing to the corresponding version of Love’s inequality, the first bounding inequality is obtained. Further, the lemmas those given in Section 2 are employed to obtain remaining two bounding inequalities.

MSC:

26A33 Fractional derivatives and integrals
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
26D15 Inequalities for sums, series and integrals
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
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