Functional inequalities for the Fox-Wright functions

In this paper,our aim is to establish some mean value inequalities for the Fox–Wright functions,suchasTurán-type inequalities, Lazarevic and Wilker-type inequalities. As applications we derive some new type inequalities for hypergeometric functions and thefour-parametric Mittag–Lefflerfunctions.Furthermore,weprovethemonotonicity of ratios for sections of series of Fox–Wright function

A note on Turán type and mean inequalities for the Kummer function

AbstractTurán-type inequalities for combinations of Kummer functions involving Φ(a±ν,c±ν,x) and Φ(a,c±ν,x) have been recently investigated in [Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (3) (2008) 279–293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007) 1–9]. In the current paper, we resolve the corresponding Turán-type and closely related mean inequalities for the additional case involving Φ(a±ν,c,x). The application to modeling credit risk is also summarized

When does a hypergeometric function pFq belong to the Laguerre–Pólya class LP⁺?

I show that a hypergeometric function p q F (a1, . . . , ap; b1, . . . , bq; ·) with p ≤ q belongs to the Laguerre–P´olya class LP + for arbitrarily large bp+1, . . . , bq > 0 if and only if, after a possible reordering, the differences ai −bi are nonnegative integers. This result arises as an easy corollary of the case p = q proven two decades ago by Ki and Kim. I also give explicit examples for the case 1 2 F

Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by $L^1$-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated $L^1$-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties holds (which is very different to $L^1$-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Tur\'{a}n's inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Tur\'{a}n type inequalities and transformations into more convenient orthogonal polynomial systems.Comment: Main changes towards first version: The part on associated symmetric Pollaczek polynomials was extended (with more emphasis on Tur\'{a}n's inequality and including a larger parameter region), and the part on little $q$-Legendre polynomials became a separate paper. We added several references and corrected a few typos. Title, abstract and MSC class were change

Generalized Volterra functions, its integral representations and applications to the Mathieu-type series

In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Fox-Wright H-functions and Meijer G-functions. From positivity conditions on the weight in these representations, we found sufficient conditions on parameters of the generalized Volterra function to prove its complete monotonicit

On some Stochastic Control Problems arising in Environmental Economics and Commodity Markets

Koch T. On some Stochastic Control Problems arising in Environmental Economics and Commodity Markets. Bielefeld: Universität Bielefeld; 2020