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Functional inequalities involving modified Struve functions

In this paper our aim is to prove some monotonicity and convexity results for the modified Struve function of the second kind by using its integral representation. Moreover, as consequences of these results, we present some functional inequalities (like Tur\'an type inequalities) as well as lower and upper bounds for modified Struve function of the second kind and its logarithmic derivative.


Introduction
In the last decades many functional inequalities and monotonicity properties for special functions (like Bessel, modified Bessel, Gaussian hypergeometric, Kummer hypergeometric) and their combinations have been established by researchers, motivated by several problems that arise in wave mechanics, fluid mechanics, electrical engineering, quantum billiards, biophysics, mathematical physics, finite elasticity, probability and statistics, special relativity and radar signal processing. Although the inequalities involving the quotients of modified Bessel functions of the first and second kind are interesting in their own right, recently the lower and upper bounds for such quotients have received increasing attention, since they play an important role in various problems of mathematical physics and electrical engineering. For more details, see for example [2] and the references therein. The modified Struve functions of the first and second kind are related to modified Bessel functions of the first kind, and thus their properties can be useful in problems of mathematical physics. In [4] Joshi and Nalwaya presented some two-sided inequalities for modified Struve functions of the first kind and for their ratios. They also deduced some Turán and Wronski type inequalities for modified Struve functions of the first kind by using a generalized hypergeometric function representation of the Cauchy product of two modified Struve functions of the first kind. Motivated by the above results, by using a known result on the monotonicity of quotients of MacLaurin series, recently in [3] we proved some monotonicity and convexity results for the modified Struve functions of the first kind. Moreover, as consequences of these results, we presented some functional inequalities as well as lower and upper bounds for modified Struve functions of the first kind. In this paper our aim is to continue the study from [3], but for the modified Struve functions of the second kind. The key tools in the proofs of the main results are the techniques developed in the extensive study of modified Bessel functions of the first and second kind and their ratios. The difficulty in the study of the modified Struve function consists in the fact that the modified Struve differential equation is not homogeneous, however, as we can see below, the integral representation of modified Struve function of the second kind is very useful in order to study its monotonicity and convexity properties.

Modified Struve function: Monotonicity patterns and functional inequalities
The modified Struve functions of the first and second kind, L ν and M ν are particular solutions of the modified Struve equation [7, p. 288] (2.1) The modified Struve function of the second kind has the power series representation where I ν stands for the modified Bessel function of the first kind.
Thus, the functions x → M ν (x) and ν → M ν (x) are indeed completely monotonic and consequently are log-convex, since every completely monotonic function is log-convex, see [11, p. 167]. Alternatively, the log-convexity of these functions can be proved also by using (2.4) and the Hölder-Rogers inequality for integrals. For inequality (2.5) just observe that M ν is decreasing on (0, ∞) for all ν > − 1 2 , and thus On the other hand, observe that x → x ν is completely monotonic on (0, ∞) for all ν ≤ 0. Thus, by part a of the theorem the function x → −M ν (x), as a product of two completely monotonic functions, is completely monotonic and log-convex on (0, ∞) for all ν ∈ − 1 2 , 0 . Now, since (see [7, p. 254] and [7, p. 291]) the function x → −M − 1 2 (x) is completely monotonic and log-convex as the product of the completely monotonic and log-convex functions x → 2 π x − 1 2 and x → e −x . Now, focus on the Turán type inequality (2.6).

Further results
In this section we give a set of other functional inequalities for the function x → M ν (x).

Theorem 2. The following inequalities hold true:
a. For all x, y > 0 and ν > − 1 2 we have b. For all ν ≥ 1 2 and x > 0 it is Moreover, the above inequality is reversed when |ν| < 1 2 and x > 0. c. For all ν ≥ 3 2 and x > 0, we have For all for all ν > −1 and x > 0, we have Proof. a. From (2.5) and part a of Theorem 1 it is clear that the function x → Γ(ν+1) (0, ∞) into (0, 1) and it is completely monotonic on (0, ∞) for all ν > − 1 2 . On the other hand, according to Kimberling [5] if a function f, defined on (0, ∞), is continuous and completely monotonic and maps (0, ∞) into (0, 1), then log f is super-additive, that is for all x, y > 0 we have Therefore we conclude the asserted inequality (3.1).

Remark 1. By (2.3) and (3.3) immediately follows the inequality
the validity range is ν ≥ 3 2 and x > 0, while the inequality is reversed when ν ∈ 1 2 , 3 2 and x > 0. Next we derive inequalities similar to (2.8) when ν ∈ − 1 2 , 0 . We note that the left-hand side of (3.6) is weaker than the left-hand side of (2.8), however, the right-hand side of (3.6) is better than the right-hand side of (2.8) Theorem 3. For all x > 0 and ν ∈ − 1 2 , 0 we have Moreover, for x > 0 and ν > 1 2 we have Proof. By c of Theorem 1 we have for all x > 0 and ν ∈ − 1 2 , 0 . On the other hand, recall that the modified Struve function M ν is a particular solution of the modified Struve equation (2.1) and consequently Combining this equation with the above inequality we get Here we used the fact that M ν (x) < 0 for x > 0 and ν ≥ − 1 2 . From the above inequality we deduce (3.6), for all x > 0 and ν ∈ − 1 2 , 0 . Moreover, since M ′ ν (x) > 0 for x > 0 and ν ∈ − 1 2 , 0 , the expression xM ′ ν (x) M ν (x) −1 is negative, which implies the right-hand side of (3.6).
It remains to prove (3.7). By using (2.12) and (2.14) we have Thus, by using (2.14), (2.12), (2.7) and the fact that M ν (x) < 0 for x > 0 and ν > − 1 2 , we have Combining this with the right-hand side of the Turán type inequality (2.6) we obtain the desired bound. Now, in order to establish a bilateral functional inequality for M ν , we need the Fox-Wright generalized hypergeometric function p Ψ q (·), with p numerator and q denominator parameters, defined by Here z, a l , b j ∈ C, α l , β j ∈ R for l ∈ {1, . . . , p} and j ∈ {1, . . . , q}. The series (3.9) converges absolutely and uniformly for all bounded |z|, z ∈ C when We note that the next inequality complements and improve (2.5).