Functional inequalities and monotonicity results for modified Lommel functions of the first kind

We establish some monotonicity results and functional inequalities for modified Lommel functions of the first kind. In particular, we obtain new Tur\'{a}n type inequalities and bounds for ratios of modified Lommel functions of the first kind, as well as the function itself. These results complement and in some cases improve on existing results, and also generalise a number of the results from the literature on monotonicity patterns and functional inequalities for the modified Struve function of the first kind.


Introduction
The modified Lommel function of the first kind t µ,ν (x) is a particular solution to the modified Lommel differential equation [13]. Modified lommel functions arise in scattering amplitudes in quantum optics [24], stress distributions in cylindrical objects [22] and the physics of two-dimensional diffusions [25] and heat conduction [8]. The modified Lommel function t µ,ν (x) generalises the modified Strive function of the first kind L ν (x) (see Section 2), which also arises in manifold applications; see [3] for a list of application areas.
Over the last several decades an extensive literature has built up on monotonicity results and functional inequalities for Bessel, modified Bessel and related functions motivated through problems in the applied sciences (see, for example, [1,21] and references therein); however, only recently have such results started to be obtained for modified Lommel functions. In [17] some monotonicity properties and convexity results for the modified Lommel function of the first kind t µ,ν (x) were obtained, from which some Turán type inequalities followed. A Redheffer type bound for the function t µ−1/2,1/2 (x) was also obtained by [17], and [9] established accurate bounds for t µ,ν (x), although these bounds only hold for 0 < x < 1. In a very recent work, [12] extended results of [10] concerning functional inequalities for modified Struve functions of the first kind to obtain bounds for the important quantities t µ,ν (x)/t µ−1,ν−1 (x), xt ′ µ,ν (x)/t µ,ν (x), t µ,ν (x)/t µ,ν (y) and the function t µ,ν (x) itself in terms of analogous expressions involving the modified Bessel functions of the first kind I ν (x). These results are quite powerful because there is substantial literature on functional inequalities for modified Bessel functions from which one can draw suitable bounds (see [1,10,21] and references therein).
Our aim in this paper is to further contribute to the recent literature on functional inequalities and monotonicity properties of modified Lommel functions of the first kind, as well as to generalise existing results for the modified Struve function of the first kind. In Section 3.1, we generalise results from the comprehensive study of monotonicity properties and functional inequalities for the modified Struve function L ν (x) given by [4], which in turn complemented and improved results of [14]. As the functions t µ,ν (x) and L ν (x) share a similar power series representation (see Section 2 for these and further properties), the approach of [4], which involves appealing to general results on the monotonicity of quotients of power series, is also effective for our purpose of modified Lommel functions of the first kind, and we are also able to obtain some other monotonicity results and functional inequalities, which complement results of [12]. We note that a few of the monotonicity and convexity results of [4] have already been generalised by [17]. In addition, in Section 3.2, we obtain new Turán type inequalities for the modified Lommel function of the first kind, which complement a different type of Turán type inequalities for t µ,ν (x) that were obtained by [17]. One of our Turán type inequalities generalises one of [4] given for the modified Struve function L ν (x), whilst our other two-sided inequality gives a new Turán type inequality in the special case of the modified Struve function L ν (x), and therefore complements the results of [5] and [6]. We also note that our Turán type inequalities naturally complement those of [2] that were given for a certain type of Lommel function of the first kind.

The modified Lommel function of the first kind
The modified Lommel function of the first kind t µ,ν (x) is defined by the hypergeometric series and arises as a particular solution of the modified Lommel differential equation [23,20] (2.1) In the literature different notation is used for the modified Lommel functions; we use the notation of [27]. The terminology modified Lommel function of the first kind is also not standard, but has recently been introduced by [12]. We shall follow [12] and use the following normalization which will be useful for our purposes, as it will remove a number of multiplicative constants from our calculations: To ease the exposition, we will also refer tot µ,ν (x) as the modified Lommel function of the first kind. From now on, we shall work with the functiont µ,ν (x); results for t µ,ν (x) can be easily inferred. We note the important special casẽ where L ν (x) is a modified Struve function of the first kind.
For x > 0, the functiont µ,ν (x) is positive if µ − ν ≥ −3 and µ + ν ≥ −3 (equivalently µ ≥ −3 and |ν| ≤ µ + 3). The functiont µ,ν (x) satisfies the following recurrence relations [12] and differentiation formula [11]: . (2.6) It will be also useful to follow [12] and introduce the following function: The functiont µ,ν (x) has the following asymptotic properties [12]: We end this section by recording that the modified Bessel function of the first kind and modified Struve function of the first kind are defined by the power series [18] . (2.10) The functions I ν (x) and L ν (x) have the following asymptotic behaviour: 3 Main results and proofs

Monotonicity results and associated inequalities
In the sequel we shall need the following result (see [7,19]).
Suppose the power series f (x) = n≥0 a n x n and g(x) = n≥0 b n x n , where a n ∈ R and b n > 0 for all n ≥ 0, both converge on (−r, r), r > 0.
For our purposes, it is important to note that Lemma 3.1 also holds when both the power series f (x) and g(x) are even, or both are odd functions.
(iv) This is very similar to part (i) of Theorem 2.1 of [17], which is given for a different normalization of the modified Lommel function t µ,ν (x). Due to the different normalization used, our result has a different range of validity. We omit the details.
(vi) By part (iv), we have which by the quotient rule can be seen to be equivalent to Using the differentiation formula (2.5) we can express this inequality in the form .
By inequality (3.17) we have that the right-hand side is positive, proving the assertion.

Remark 3.3. (i) Parts (iii)-(vi) generalise monotonicity results for the modified Struve
function L ν (x) given in Theorem 2.2 of [4]. Indeed, the results exactly reduce to those of [4] in the case µ = ν.
(ii) Inequality (3.16) (and its reverse) complement the following two-sided inequality of [12], where and both the lower and upper bounds are valid for µ > −2, −1 < ν < µ + 1. Using the limiting forms (2.7) and (2.11) we see that inequality (3.16) (and its reverse) and the upper bound of (3.21) are tight in the limit x ↓ 0, but that the lower bound of (3.21) is not. All bounds are of the correct asymptotic order O(x −1/2 e x ), as x → ∞, but only the lower bound of (3.21) is tight in this limit (see (2.8) and (2.13)). It is interesting to note that inequality (3.16) is expressed in terms of I µ+1 (x), whereas (3.21) is expressed in terms of I ν (x). The modified Bessel function I µ+1 (x) has the same asymptotic order ast µ,ν (x) in both the limits x ↓ 0 and x → ∞, which helps to explain how the bound (3.16) has a similar performance to (3.21) despite taking a simpler form.
We also note that inequalities (3.15) and (3.16) are very useful in that they allow one to obtain a number of different bounds fort µ,ν (x) as a consequence of bounds in the existing literature for L µ (x) and I µ+1 (x). For reasons of brevity, we only note one such example. Applying inequality (4.59) of [12] to inequality (3.15) yields the following neat bound which further complements inequalities (3.16) and (3.21): for This inequality is also tight as x ↓ 0 and has the correct asymptotic order as x → ∞.
(iii) A similar upper bound for the condition number |xt ′ µ,ν (x)/t µ,ν (x)| was obtained in inequality (4.46) of [12]: for x > 0, An asymptotic analysis of the bounds using the limiting forms (2.7) and (2.8) shows that our bound (3.19) outperforms the bound of [12] in the limit x ↓ 0, whilst the reverse is true as x → ∞.
This inequality is sharp in the limit x ↓ 0. Note that inequality (3.24) also follows as a special case of inequality (3.17). Inequalities (3.23) and (3.24) generalise bounds of [4] involving the modified Struve function L ν (x). The bounds of [4] improved the range of validity of earlier results of [14]. A number of inequalities for the quantitiest µ,ν (x)/t µ,ν (y) andt µ,ν (x)/t µ−1,ν−1 (x) were obtained by [12]. The simple bounds obtained in this remark have the advantage over those of [12] by having a larger range of validity. Also, unlike inequality (3.24), none of the upper bounds of [12] fort µ,ν (x)/t µ−1,ν−1 (x) are sharp in the limit x ↓ 0. However, the bounds of [12] perform much better for 'large' x than the bounds given in this remark.
We now obtain a further monotonicity result and associated inequality that complements an inequality of [12].
We will need the following lemma [15,Remark 3].