Turán Type Inequalities for Tricomi Confluent Hypergeometric Functions

Abstract

Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher-Snedecor F distribution. Moreover, some complete monotonicity results are given concerning Turán determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia (in Constr. Approx. 26:1-9, 2007).

Universal bounds and monotonicity properties of ratios of Hermite and parabolic cylinder functions

“…Notice that the validity of (1.5) was first proved by G. Szegö in 1948 (see [22]). Since then, inequalities of this form have attracted a lot of attention, and have been proved to be valid for other polynomials such as Hermite (obtained from Hermite functions by taking ν ∈ N), Jacobi, Laguerre or ultraspherical polynomials (see [12,22], among others), and for special functions as (modified) Bessel, Gamma, parabolic cylinder or hypergeometric functions (see [2,4,5,6,7,23], among many others). Applications of Turán type inequalities can be found in many fields, ranging from biophysics (see [3] and the references therein) to information theory (see [15]) and stochastic control (see [8,11]).…”

Turán type inequalities for confluent hypergeometric functions of the second kind

“…where a > 0, c < a + 2, x > 0 in the first inequality, and a > 1, c < a + 1, x > 0 in the second inequality. Next, using some ideas from [3, Theorem 2] and the Turán type inequalities for confluent hypergeometric functions of the second kind [8], we deduce some new inequalities for confluent hypergeometric functions of the second kind.…”

“…In [7], the author found some tight bounds for Turánian of modified Bessel functions of first and second kind. Motivated by the results from [7], in this paper we find tight bounds for the Turánians of confluent hypergeometric functions of the second kind and we offer some alternative proofs of the Turán type inequalities given in [8]. Moreover, by using a technique similar to [3], we derive some new inequalities for confluent hypergeometric functions of the second kind.…”

“…Recently Baricz and Ismail [8] have deduced some Turán type inequalities for confluent hypergeometric functions of the second kind. In [7], the author found some tight bounds for Turánian of modified Bessel functions of first and second kind.…”

Optimal liquidation under stochastic liquidity

“…Note that such quotients of special functions are related to so called Turan-type inequalities, cf. [BI13].…”