Turán type inequalities for Struve functions

Abstract

Abstract. Some Turán type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like Mittag-Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modified Struve functions of the first and second kind. Turán type inequalities for Struve functions of the first kindLet us start with a well-known relation between Bessel functions of the first kind J ν and Struve functions of the first kind H ν . Namely, for all n ∈ {0, 1, . . . } and x ∈ R we have [11, p. 291]. Now, let us recall the Turán type inequality for Bessel functions of the first kind, that is,, where x ∈ R and ν > −1. Combining the above relation with (1.1) we obtain the Turán type inequality, which holds for all x ∈ R and ν ∈ − 1 2 , − 3 2 , . . . . Moreover, since for x ∈ R and ν ≥ 0 the Turán type inequality (1.1) can be improved asit follows that the inequality (1.2) can be improved too aswhich holds for all x ∈ R and ν ∈ − 1 2 , − 3 2 , . . . . For more details on the above Turán type inequalities for Bessel functions of the first kind we refer to the papers [12,14,15,16] and also to the survey paper [2]. Taking into account the above inequalities it is natural to ask whether the Turán type inequalities (1.2) and/or (1.3) hold true for other values of ν. In this paper we will concentrate on this problem and we present some interesting results concerning Turán type inequalities for Struve functions of the first and second kind. As we can see below the analysis of Struve functions is somewhat more complicated than that of Bessel functions, however, its nature is quite similar for some values of ν. This section is devoted to Turán type inequalities for Struve functions of the first kind, while the next section contains some results, like Turán type inequalities and complete monotonicity results on Struve functions of the second kind. Before we present the main results of this section we first show some preliminary results which will be used in the sequel and which may be of independent interest. Since Struve functions are frequently used in many places in physics and applied mathematics, we believe that our results may be useful for other scientists interested in Struve functions. We also note that the analogous results for modified Struve functions of the first and second kind were already deduced by Baricz and Pogány [3,4] by using the techniques developed in the case of modified Bessel functions of the first and second kind. Moreover, the results presented in this section complement naturally the known results for a particular case of Lommel functions of the first kind, obtained recently by Baricz and...

Bounds for modified Struve functions of the first kind and their ratios

“…Recently, [8] used a classical result on the monotonicity of quotients of Maclaurin series and techniques developed in the extensive study of modified Bessel functions and their ratios to obtain monotonicity results and, as a consequence, functional inequalities for the modified Struve function of the first kind L ν (x) that complement and improve the results of [22]. Further results and a new proof of a Turán-type inequality for the modified Struve function of the first kind are given in [6], and monotonicity results and functional inequalities for the modified Struve function of the second kind M ν (x) = L ν (x) − I ν (x) are given in [9]. It should be noted that the techniques used in [8] and [9] to obtain functional inequalities for L ν (x) and M ν (x) are quite different (this is also commented on in [12]), which is in contrast to the literature on modified Bessel functions in which functional inequalities for I ν (x) and K ν (x) are often developed in parallel.…”

Inequalities for series in q-shifted factorials and q-gamma functions

“…Further examples of (1) can be given with f n not expressible in terms of gamma functions. Our papers [8,9,10,12,13,14,15] cover nearly all possible combinations of A 0 and A 1 from the collection (2). Most our results are shaped as follows.…”

“…, Γ(a + ·) Γ(b + ·) (2) and f n is a (usually non-negative) real sequence. Here Γ stands for Euler's gamma function and a, b are non-negative parameters.…”

Inequalities for Integrals of the Modified Struve Function of the First Kind II

“…2 ) − (2ν + n + 1)x n+2 √ π2 ν+n+1 (n + 1)(n + 2)Γ(ν + n + 5 2 ) = (2(ν + n + 1)(n + 2) − (2ν + n + 1))x n+2 √ π2 ν+n+1 (n + 1)(n + 2)Γ(ν + n + 5 2 ) = 2(n + 1)(ν + n + 3 2 )x n+2 √ π2 ν+n+1 (n + 1)(n + 2)Γ(ν + n + 5 2 ) = x n+2 √ π2 ν+n (n + 2)Γ(ν + n + 3 2 )…”