Carla Giordano

Unified treatment of Gautschi-Kershaw type inequalities for the gamma function

Abstract Gautschi, Kershaw, Lorch, Laforgia and other authors gave several inequalities for the ratio Γ(x + 1) Γ(x + s) where, as usual, Γ denotes the gamma function. In this paper we give a unified treatment of all their results and prove, among other things, new inequalities for the above ratio, which involve the psi function. Inequalities for the ratio of two gamma functions are useful, for example, to deduce Bernstein-type inequalities for ultraspherical polynomials. We give an example of this type.

On the Bernstein-type inequalities for ultraspherical polynomials

We present a survey of the most recent results and inequalities for the gamma function and the ratio of the gamma functions and study, among other things, the relation between these results and known inequalities for ultraspherical polynomials. In particular, we discuss the inequality (sin θ)λ|Pn(λ)(cos θ)| < 21-λ/Γ(λ) Γ(n + 3/2λ)/Γ(n + 1 + 1/2λ), 0 ≤ θ ≤ π, where Pn(λ)(cos θ) denotes the ultraspherical polynomial of degree n, established by Alzer (Arch. Math. 69 (1997) 487) and the one established by Durand (In: R.A. Askey (Ed.), Theory and Application of Special Functions, Proceedings of the Advanced Seminar on Mathematical Research Center, University of Wisconsin, Madison, Vol. 35, Academic Press, New York, 1975, p. 353) (sin θ)λ|Pn(λ)(cos θ)| ≤ Γ(n/2 + λ)/Γ(λ)Γ(n/2 + 1), 0 ≤ θ ≤ π.

Luigi Gatteschi's work on asymptotics of special functions and their zeros

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.

Elementary approximations for zeros of Bessel functions

Let jvk, yvk and cvk denote the kth positive zeros of the Bessel functions Jv(x), Yv(x) and of the general cylinder function Cv(x) = cos ?Jv(x)?sin ?Yv(x), 0 ? ? < ?, respectively. In this paper we extend to cvk, k = 2, 3,..., some linear inequalities presently known only for jvk. In the case of the zeros yvk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero jv1 we compare the linear enequalities given in 9] with some other known inequalities.

Error bounds for mcmahon's asympotic approximations of the zeros of the bessel functions

We derive bounds for the error term in the McMahons asymptotic approximation of the positive zeroes Bessel function . The cases and for all the zeros exceeding are discussed.

Convexity andq-gamma function

Inequalities and convexity properties known for the gamma function are extended to theq-gamma function, 0<q<1. Applying some classical inequalities for convex functions, we deduce monotonicity results for several functions involving theq-gamma function. Further applications to the probability theory are given.

Further monotonicity and convexity properties of the zeros of cylinder functions

Let cvk be the kth positive zero of the cylinder function Cv(x,α)=Jv(x) cos α−Yv sin α, 0⩽α<π, where Jv(x) and Yv(x) are the Bessel functions of the first and the second kind, respectively. We prove that the function v(d2cvkddv2+δ)cvk increases with v⩾0 for suitable values of δ and k−απ⩾ 0.7070… . From this result under the same conditions we deduce, among other things, that cvk+12δv2 is convex as a function of v⩾0. Moreover, we show some monotonicity properties of the function c2vkv. Our results improve known results.

Inequalities for Some Special Functions and their Zeros

We establish inequalities for the Bessel functions Jν (x) of the first kind, by means of the arithmetic geometric mean inequality and the Infinite product formula for Jν(x). A concavity property is also obtained for the positive zeros jνk(k = 1, 2, …) of Jν(x) using a lower bound for the second derivative of recently established in [3]. Finally we show a monotonicity property of the zeros of Legendre polynomials. This property is proved as a consequence of the classical Sturm comparison theorem.