Functional inequalities for modified Struve functions

Abstract

Abstract. In this paper, by using a general result on the monotonicity of quotients of power series, our aim is to prove some monotonicity and convexity results for the modified Struve functions. Moreover, as consequences of the above mentioned results, we present some functional inequalities as well as lower and upper bounds for modified Struve functions. Our main results complement and improve the results of Joshi and Nalwaya [11].

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

“…In this paper, we obtain new bounds for the ratios L ν (x)/L ν−1 (x) and L ν (x)/L ν (y), the condition numbers xL ′ ν (x)/L ν (x) and the modified Struve function L ν (x) itself. These results complement and, at least in some cases, improve those given in [8,22]. Our approach is quite different, though.…”

“…where we have equality if and only if ν = 1 2 . It should also be noted that a more complicated bound, valid for ν ≥ 3 2 , which improves on (2.25) is given by inequality (3.1) of [8]. Let us compare our bound (2.23) with (2.25).…”

“…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. For this reason, in this paper, with the exception of Remark 2.5, we restrict our attention to the modified Struve function L ν (x).…”

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

“…Setting µ = ν in (4.54) yields the upper boundL ν (x) < Γ(ν + 1) 3(2ν + 3) √ πΓ(ν + 3 xI ν (x) x 2 + 3(2ν + 3) , x > 0, ν > −1,(4.59)which complements the following inequality of[9]:L ν (x) ≤ 2Γ(ν + 2) √ πΓ(ν + 3 2 ) I ν+1 (x), x > 0, ν ≥ − 1 2 , (4.60)…”

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

“…As already noted, the reason for this similarity is because many of the properties of I ν (x) that were exploited in the proofs of [8,10,13] are shared by L ν (x), which we now list. All these formulas can be found in [17], except for the inequality which is given in [2]. Further inequalities for L ν (x) can be found in [2,3,12,15], some of which improve on the inequality of [2].…”