Bounds for an integral involving the modified Struve function of the first kind
aa r X i v : . [ m a t h . C A ] J a n BOUNDS FOR AN INTEGRAL INVOLVING THE MODIFIEDSTRUVE FUNCTION OF THE FIRST KIND
ROBERT E. GAUNT
Abstract.
Simple upper and lower bounds are established for the integral R x e − βt t ν L ν ( t ) d t , where x > ν > −
1, 0 < β < L ν ( x ) is the modifiedStruve function of the first kind. These bounds complement and improve onexisting results, through either sharper bounds or increased ranges of validity.In deriving our bounds, we obtain some monotonicity results and inequalitiesfor products of the modified Struve function of the first kind and the modifiedBessel function of the second kind K ν ( x ), as well as a new bound for the ratio L ν ( x ) / L ν − ( x ). Introduction
In a series of recent papers [12, 14, 18], simple upper and lower bounds, involvingthe modified Bessel function of the first kind I ν ( x ), were established for the integral(1.1) Z x e − βt t ν I ν ( t ) d t, where x >
0, 0 ≤ β <
1. The conditions imposed on ν differed for several ofthe inequalities, although in all cases ν > − , which ensures that the integralexists. For 0 < β < Q -function, which arises in radarsignal processing.The modified Struve function of the first kind is defined, for x ∈ R and ν ∈ R ,by the power series L ν ( x ) = ∞ X k =0 (cid:0) x (cid:1) ν +2 k +1 Γ( k + )Γ( k + ν + ) . The modified Struve function L ν ( x ) is closely related to the modified Bessel function I ν ( x ), either sharing or having close analogues to the properties of I ν ( x ) that wereused by [12, 14, 18] to derive inequalities for the integral (1.1). The function L ν ( x ) is itself a widely used special function, with numerous applications in theapplied sciences, such as perturbation approximations of lee waves in a stratifiedflow [24], leakage inductance in transformer windings [22], and quantum-statistical Date : 27 January 2021.2010
Mathematics Subject Classification.
Primary 33C20; 26D15.The author is supported by a Dame Kathleen Ollerenshaw Research Fellowship. distribution functions of a hard-sphere system [26]; see [3] for examples of furtherapplication areas. Basic properties of the modified Struve function L ν ( x ) can befound in standard references, such as [28]. We collect the basic properties that willbe needed in this paper in Appendix AThe natural analogue of the problem studied by [12, 14, 18] is to ask for simpleinequalities, involving the modified Struve function of the first kind, for the integral(1.2) Z x e − βt t ν L ν ( t ) d t, where x >
0, 0 ≤ β < ν > − ν ensuring the integralexists). This problem was first studied in the recent paper [15], and will also be thesubject of this paper.The integral (1.2) can be evaluated exactly in terms of the modified Struvefunction L ν ( x ) in the case β = 1. For all ν > − and x > Z x e − t t ν L ν ( t ) d t = e − x x ν +1 ν + 1 (cid:0) L ν ( x ) + L ν +1 ( x ) (cid:1) − γ (2 ν + 2 , x ) √ π ν (2 ν + 1)Γ( ν + ) , where γ ( a, x ) = R x e − t t a − d t is the lower incomplete gamma function. This for-mula can be verified directly by a short calculation using the differentiation formula(A.2) and identity (A.1) given in Appendix A. When β = 0 the integral (1.2) cannotbe evaluated in terms of the function L ν ( x ), but an exact formula is available interms of the generalized hypergeometric function p F q (cid:0) a , . . . , a p ; b , . . . , b q ; x (cid:1) = ∞ X k =0 ( a ) k · · · ( a p ) k ( b ) k · · · ( b q ) k x k k ! , where the Pochhammer symbol is defined by ( a ) = 1 and ( a ) k = a ( a + 1)( a +2) · · · ( a + k − k ≥
1. Indeed, for − ν − / ∈ N , we have the representation L ν ( x ) = x ν +1 √ π ν Γ( ν + ) F (cid:18)
1; 32 , ν + 32 ; x (cid:19) , and by a straightforward calculation we have that, for ν > − x > Z x t ν L ν ( t ) d t = x ν +2 √ π ν +1 ( ν + 1)Γ( ν + ) F (cid:18) , ν + 1; 32 , ν + 32 , ν + 2; x (cid:19) . The integral (1.2) can also be evaluated when 0 < β <
1, but the formula is morecomplicated: for ν > − x > Z x e − βt t ν L ν ( t ) d t = ∞ X k =0 − ν − k β − k − ν − Γ( k + )Γ( k + ν + ) γ (2 k + 2 ν + 2 , βx ) . These complicated formulas provide the motivation for establishing simple bounds,involving the modified Struve function L ν ( x ) itself, for the integral (1.2).Several upper bounds and a lower bound for the integral (1.2) were establishedby [15] by adapting the techniques used by [12, 14] to bound the analogous integral(1.1) involving the modified Bessel function I ν ( x ). In this paper, we complementthe work of [15] by obtaining several lower bounds for the integral (1.2) (Theorem2.1), one of which is a strict improvement on the only lower bound given in [15].In fact, all lower bounds obtained in this paper are tight in the limit x → ∞ , afeature not seen in the lower bound of [15]. We also extend the range of validityof the upper bounds given in [15] from ν ≥ to ν > − (Theorem 2.2), with N INTEGRAL INVOLVING THE MODIFIED STRUVE FUNCTION OF THE FIRST KIND 3 our bounds taking the same functional form, but with larger numerical constants.We shall proceed in a similar manner to [15], by adapting the approach used inthe recent paper [18] to obtain similar improvements on the bounds of [12, 14]that were obtained for the related integral (1.1) involving I ν ( x ). We establish ourupper bounds by proving a series of lemmas, which may be of independent interest.Lemma 3.3 gives another upper bound for the integral (1.2), which outperformsour bounds from Theorem 2.2 for ‘large’ values of x . In Lemma 3.1, we provide anew bound for the ratio L ν ( x ) / L ν − ( x ). Lemma 3.2 gives monotonicity results andinequalities for some products involving the modified Struve function L ν ( x ) and themodified Bessel function of the second kind K ν ( x ) that complement existing resultsconcerning products involving the modified Bessel functions I ν ( x ) and K ν ( x ). Thelemmas are collected and proved in Section 3, and the main results are proved inSection 4. Elementary properties of the modified Struve function L ν ( x ) and themodified Bessel functions that are needed in the paper are collected in AppendixA. 2. Main results and comparisons
The inequalities given in the following Theorems 2.1 and 2.2 are natural ana-logues of inequalities that have been recently obtained by [18] for the related integral R x e − βt t ν I ν ( t ) d t . The inequalities also complement and improve on bounds of [15]for the integral (1.2). Theorems 2.1 and 2.2 and Proposition 2.3 below are provedin Section 4. Theorem 2.1.
Let < β < . Then, for x > , Z x e − βt t ν L ν ( t ) d t > − β (cid:26) e − βx x ν L ν ( x ) − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) , − < ν ≤ , (2.1) Z x e − βt t ν L ν ( t ) d t > − β (cid:26)(cid:18) − ν (2 ν − − β ) 1 x (cid:19) e − βx x ν L ν ( x ) − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) , ν ≥ , (2.2) Z x e − βt t ν L ν ( t ) d t > e − βx x ν ∞ X k =0 β k L ν + k +1 ( x ) , ν > − . (2.3) Inequalities (2.1)–(2.3) are tight in the limit x → ∞ . Recall that γ ( a, x ) = R x e − t t a − d t is the lower incomplete gamma function. Theorem 2.2.
Let < β < . Then, for x > , Z x e − βt t ν L ν ( t ) d t < ν + 29(2 ν + 1)(1 − β ) e − βx x ν L ν +1 ( x ) , ν > − , (2.4) Z x e − βt t ν L ν ( t ) d t < ν + 15(2 ν + 1)(1 − β ) e − βx x ν L ν ( x ) , ν > − , (2.5) ROBERT E. GAUNT Z x e − βt t ν L ν ( t ) d t > − β (cid:26)(cid:18) − ν (2 ν + 27)(2 ν − − β ) 1 x (cid:19) e − βx x ν L ν ( x ) − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) , ν > . (2.6) Inequality (2.6) is tight as x → ∞ . The inequalities in the following proposition are stronger than inequalities (2.1),(2.2) and (2.6), because L ν +1 ( x ) < L ν ( x ), x > ν ≥ − (see (A.8)). Proposition 2.3.
Let < β < . Then, for x > , Z x e − βt t ν L ν +1 ( t ) d t > − β (cid:26) e − βx x ν L ν ( x ) − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) , − < ν ≤ , (2.7) Z x e − βt t ν L ν +1 ( t ) d t > − β (cid:26)(cid:18) − ν (2 ν − − β ) 1 x (cid:19) e − βx x ν L ν ( x ) − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) , ν ≥ , (2.8) Z x e − βt t ν L ν +1 ( t ) d t > − β (cid:26)(cid:18) − ν (2 ν + 27)(2 ν − − β ) 1 x (cid:19) e − βx x ν L ν ( x ) − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) , ν > . (2.9) Remark . In this remark, we discuss the performance of our bounds given inTheorems 2.1 and 2.2, and make comparisons between our bounds and those givenby [15] for the integral (1.2). Throughout this remark 0 < β < R x e − βt t ν L ν ( t ) d t > e − βx x ν L ν +1 ( x ), x > ν > − , withthis bound in fact being the first term in the infinite series of the lower bound(2.3). The other lower bounds from Theorems 2.1 and 2.2, that is (2.1), (2.2) and(2.6), all perform worse than (2.3) and the bound of [15] for ‘small’ x . Indeed, itis easily seen that the lower bounds in (2.1) and (2.2) are negative for sufficientlysmall x , whilst a simple asymptotic analysis of the bound (2.6) using (A.4) showsthat, for − < ν <
0, the limiting form of this bound is ν ν +1 x ν +1 √ π ν Γ( ν +3 / < x ↓
0. For the case ν = 0 the bound is again negative for sufficiently small x : − β { e − βx L ( x ) − πβ (1 − e − βx ) } ∼ − βx π (1 − β ) , as x ↓
0. The bounds (2.1), (2.2)and (2.6) do, however, perform well for ‘large’ x . Unlike the bound of [15], thesebounds are tight as x → ∞ , and this is achieved without the need of an infinite suminvolving modified Struve functions of the first kind as given in the bound (2.3).Inequality (2.13) of [15] gives the following upper bound: for x > Z x e − βt t ν L ν ( t ) d t < e − βx x ν (2 ν + 1)(1 − β ) (cid:18) ν + 1) L ν +1 ( x ) − L ν +3 ( x ) − x ν +2 √ π ν +2 ( ν + 1)Γ( ν + ) (cid:19) , ν ≥ , N INTEGRAL INVOLVING THE MODIFIED STRUVE FUNCTION OF THE FIRST KIND 5 < ν + 1)(2 ν + 1)(1 − β ) e − βx x ν L ν +1 ( x ) , ν ≥ . (2.10)Another upper bound is obtained by combining inequalities (2.10) and (2.12) of[15]: for x > Z x e − βt t ν L ν ( t ) d t < − β e − βx x ν L ν ( x ) , ν ≥ . Inequalities (2.4) and (2.5) increase the range of validity of inequalities (2.10) and(2.11) to ν > − at the cost of larger multiplicative constants. These larger con-stants arise because our derivations of inequalities (2.4) and (2.5) are more involvedthan those of [15] for inequalities (2.10) and (2.11). Indeed, we arrive at our boundsby applying a series of inequalities collected in Lemmas 3.1–3.4, which when com-bined leads to a build up of errors. The reason we needed a more involved anal-ysis was because the derivations of [15] rely heavily on the use of the inequality L ν ( x ) < L ν − ( x ), which holds for x > ν ≥ (see (A.8)), and without this usefulinequality at our disposal (we have ν > − ) we required a more involved and lessdirect proof. It is worth noting that we can combine our bound (2.4) and the bound(2.10) of [15] to obtain the bound, for x > Z x e − βt t ν L ν ( t ) d t < A ν (2 ν + 1)(1 − β ) e − βx x ν L ν +1 ( x ) , ν > − , where A ν = 2( ν + 1) for ν ≥ , and A ν = 2 ν + 29 for | ν | < . A similar inequalitycan be obtained by combining our bound (2.5) and the bound (2.11) of [15].The inequalities obtained in this paper along with those presented in this re-mark allow for various double inequalities to be given for the integral (1.2). As anexample, for x > − βx x ν ∞ X k =0 β k L ν + k +1 ( x ) < Z x e − βt t ν L ν ( t ) d t < − β e − βx x ν L ν ( x ) , ν ≥ . With the aid of
Mathematica we calculated the relative error in estimating F ν,β ( x ) = R x e − βt t ν L ν ( t ) d t by the upper bound in (2.12) (denoted by U ν,β ( x )), and the lowerbound truncated at the fifth term in the sum, L ν,β ( x ) = e − βx x ν P k =0 β k L ν + k +1 ( x ).We report the results in Tables 1 and 2. For fixed x and ν , we see that increas-ing β increases the relative error in approximating F ν,β ( x ) by either L ν,β ( x ) or U ν,β ( x ). Both the lower and upper bounds in (2.12) are tight as x → ∞ , and wesee that, for fixed ν and β , the relative error in approximating F ν,β ( x ) by U ν,β ( x )decreases as x increases. However, as we have truncated the sum, L ν,β ( x ) is nottight as x → ∞ . The effect of truncating the sum is most pronounced for larger β and larger x . For β = 0 . P ∞ k =0 . k = 4 and P k =0 . k = 3 . x →∞ (cid:0) − L ν, . ( x ) F ν, . ( x ) (cid:1) = 0 . ν > − , where we also made use of the limitingforms (4.6) and (A.5). In contrast, lim x →∞ (cid:0) − L ν, . ( x ) F ν, . ( x ) (cid:1) = 9 . × − , whichis fairly negligible. The upper bound U ν,β ( x ) is of the wrong asymptotic order as x ↓ U ν,β ( x ) F ν,β ( x ) ∼ ν +1)(1 − β ) x , as x ↓ x . The lower bound L ν,β ( x ) performs better for ‘small’ x ; indeed, it isof the correct asymptotic order as x ↓ x ↓ (cid:0) − L ν,β ( x ) F ν,β ( x ) (cid:1) = ν +3 . ROBERT E. GAUNT
Table 1.
Relative error in approximating F ν,β ( x ) by L ν,β ( x ). PPPPP ( ν, β ) x , .
25) 0.2051 0.1976 0.1413 0.1028 0.0656 0.0346 0.0182(2 . , .
25) 0.1276 0.1320 0.1092 0.0863 0.0591 0.0329 0.0177(5 , .
25) 0.0781 0.0831 0.0773 0.0670 0.0503 0.0302 0.0169(10 , .
25) 0.0439 0.0465 0.0468 0.0444 0.0378 0.0257 0.0155(1 , .
5) 0.2111 0.2582 0.2259 0.1843 0.1341 0.0870 0.0602(2 . , .
5) 0.1304 0.1635 0.1606 0.1426 0.1133 0.0791 0.0570(5 , .
5) 0.0793 0.0971 0.1039 0.1004 0.0881 0.0680 0.0522(10 , .
5) 0.0443 0.0514 0.0569 0.0590 0.0580 0.0515 0.0440(1 , .
75) 0.2171 0.3359 0.3723 0.3659 0.3369 0.2953 0.2683(2 . , .
75) 0.1333 0.2036 0.2458 0.2597 0.2640 0.2581 0.2500(5 , .
75) 0.0805 0.1142 0.1446 0.1635 0.1850 0.2084 0.2226(10 , .
75) 0.0447 0.0569 0.0705 0.0825 0.1028 0.1400 0.1774
Table 2.
Relative error in approximating F ν,β ( x ) by U ν,β ( x ). PPPPP ( ν, β ) x , .
25) 9.4597 0.3208 0.0888 0.0521 0.0292 0.0139 0.0068(2 . , .
25) 17.4185 0.9887 0.3593 0.2156 0.1197 0.0565 0.0274(5 , .
25) 30.7218 2.1879 0.8593 0.5134 0.2806 0.1300 0.0625(10 , .
25) 57.3655 4.7301 1.9918 1.1901 0.6378 0.2868 0.1351(1 , .
5) 14.2938 0.5538 0.1530 0.0839 0.0452 0.0212 0.0103(2 . , .
5) 26.1923 1.5400 0.5661 0.3363 0.1842 0.0858 0.0414(5 , .
5) 46.1220 3.3214 1.3161 0.7868 0.4286 0.1972 0.0943(10 , .
5) 86.0701 7.1185 3.0084 1.8015 0.9664 0.4339 0.2037(1 , .
75) 28.8028 1.3243 0.4124 0.2137 0.1021 0.0444 0.0210(2 . , .
75) 52.5169 3.2293 1.2300 0.7305 0.3933 0.1783 0.0845(5 , .
75) 92.3236 6.7374 2.7112 1.6308 0.8892 0.4056 0.1918(10 , .
75) 172.1854 14.2887 6.0686 3.6482 1.9648 0.8827 0.4126 Lemmas
We prove Theorem 2.2 through the following series of lemmas, which may be ofindependent interest.
Lemma 3.1.
Let ν > and x > . Then L ν ( x ) L ν − ( x ) > x ν + 1 + x . (3.1) This bound is tight in the limits x ↓ and x → ∞ . Lemma 3.2.
Suppose ν ≥ − . Then the functions x K ν +1 ( x ) L ν ( x ) and x xK ν +2 ( x ) L ν ( x ) are strictly decreasing on (0 , ∞ ) . As a consequence of the lattermonotonicity result, we have the following tight two-sided inequality: (3.2) 12 < xK ν +2 ( x ) L ν ( x ) < ν + 2) √ π Γ( ν + ) , x > . We also have that, for x > , xK ν +1 ( x ) L ν ( x ) < , (3.3) xK ν +3 ( x ) L ν ( x ) < ν + 2) √ π Γ( ν + ) (cid:18) ν + 5 x (cid:19) . (3.4) N INTEGRAL INVOLVING THE MODIFIED STRUVE FUNCTION OF THE FIRST KIND 7
Suppose now that − ≤ ν ≤ . Then, for x > , xK ν +2 ( x ) L ν ( x ) < , (3.5) xK ν +3 ( x ) L ν ( x ) <
32 + 9 x , (3.6) xK ν +3 ( x ) L ν +1 ( x ) < . (3.7) Lemma 3.3.
Let ν > − and < β < . Fix x ∗ > − β . Then, for x ≥ x ∗ , (3.8) Z x e − βt t ν L ν ( t ) d t < M ν,β ( x ∗ )e − βx x ν L ν +1 ( x ) , where (3.9) M ν,β ( x ∗ ) = max (cid:26) ν + 3 + 2 x ∗ ν + 1 , x ∗ (1 − β ) x ∗ − (cid:27) . Lemma 3.4.
Suppose that − < ν ≤ and < β < . Then, for x > , e βx K ν +3 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t < ν + 1)(1 − β ) , (3.10) e βx K ν +2 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t < ν + 1)(1 − β ) . (3.11) Remark . The monotonicity results of Lemma 3.2 for the products K ν +1 ( x ) L ν ( x )and xK ν +2 ( x ) L ν ( x ) complement monotonicity results that have been establishedfor the products K ν ( x ) I ν ( x ) (see [1, 2, 29, 30]), xK ν ( x ) I ν ( x ) (see [21]) and xK ν +1 ( x ) I ν ( x )(see [13]). We also note that a number of bounds for the product K ν ( x ) I ν ( x ) havebeen obtained by [4]. In light of these results, it is natural to ask whether a mono-tonicity result is available for the product xK ν +1 ( x ) L ν ( x ), which is also present inLemma 3.2. It turns out that, for fixed ν > − , xK ν +1 ( x ) L ν ( x ) is not a monotonefunction of x on (0 , ∞ ). Indeed, applying the limiting forms (A.4)–(A.7) gives that xK ν +1 ( x ) L ν ( x ) ∼ Γ( ν + 1) x √ π Γ( ν + ) , x ↓ ,xK ν +1 ( x ) L ν ( x ) ∼
12 + 2 ν + 14 x , x → ∞ , which tells us that xK ν +1 ( x ) L ν ( x ) is an increasing function of x for ‘small’ x anda decreasing function of x for ‘large’ x if ν > − . Remark . Inequality (3.8) of Lemma 3.3 is more accurate than inequalities (2.4)and (2.5) of Theorem 2.1 for ‘large’ x . As an example, applying Lemma 3.3 with x ∗ = − β gives that, for x ≥ − β , ν > − , 0 < β < Z x e − βt t ν L ν ( t ) d t < ν + 1 (cid:18) ν + 3 + 41 − β (cid:19) e − βx x ν L ν +1 ( x ) , which is an improvement on both (2.4) and (2.5) in its range of validity. Proof of Lemma 3.1.
We begin by noting the following bound of [16, Theorem 2.2]: L ν ( x ) L ν − ( x ) > (cid:18) I ν − ( x ) I ν ( x ) + 2 b ν ( x ) x (cid:19) − , x > , ν ≥ , ROBERT E. GAUNT where b ν ( x ) = ( x/ ν +1 √ π Γ( ν +3 / L ν ( x ) . Part (iii) of Lemma 2.1 of [16] tells us that b ν ( x ) < for all x > ν ≥
0, and so we have the simpler bound(3.12) L ν ( x ) L ν − ( x ) > (cid:18) I ν − ( x ) I ν ( x ) + 1 x (cid:19) − , x > , ν ≥ . The ratio of modified Bessel functions of the first kind can be bounded by theinequality I ν ( x ) I ν − ( x ) > x ν + x , x > , ν > , which is the simplest lower bound in a sequence of rational bounds obtained by[25]. Applying this bound to (3.12) then gives us our desired bound (3.1). Finally,the assertion that the bound is tight in the limits x ↓ x → ∞ follow easilyfrom an application of the limiting forms (A.4) and (A.5) and the standard formulaΓ( x + 1) = x Γ( x ). (cid:3) Proof of Lemma 3.2. (i) Note that we can write K ν +1 ( x ) L ν ( x ) = f ν ( x ) g ν ( x ), where f ν ( x ) = K ν +1 ( x ) I ν +1 ( x ) and g ν ( x ) = L ν ( x ) /I ν +1 ( x ). It has been shown that, for ν > − f ν ( x ) is a strictly decreasing function of x on (0 , ∞ ) (see [2], which extendsthe range of validity of results of [1, 29]), and part (i) of Theorem 2.2 of [5] statesthat, for ν ≥ − , g ν ( x ) is a decreasing function of x on (0 , ∞ ). As a product oftwo strictly positive functions, one of which is strictly decreasing and the otherdecreasing, it follows that, for ν ≥ − , the function x K ν +1 ( x ) L ν ( x ) is strictlydecreasing on (0 , ∞ ).The proof that, for ν ≥ − , the function x xK ν +2 ( x ) L ν ( x ) is strictly de-creasing on (0 , ∞ ) is similar. We note that xK ν +2 ( x ) L ν ( x ) = h ν ( x ) g ν ( x ), where h ν ( x ) = xK ν +2 ( x ) I ν +1 ( x ). Lemma 3 of [13] asserts that, for ν ≥ − , f ν ( x ) is astrictly decreasing function of x on (0 , ∞ ), and the proof now proceeds exactly asthe previous one concerning the monotonicity of the function x K ν +1 ( x ) L ν ( x ).The upper and lower bounds in (3.2) now follow from using the limiting forms (A.4)–(A.7) to calculate the limits lim x ↓ xK ν +2 ( x ) L ν ( x ) and lim x →∞ xK ν +2 ( x ) L ν ( x ).(ii) Inequality (3.3) is obtained by combining the inequality L ν ( x ) < I ν ( x ), x > ν ≥ − , with the bound xK ν +1 ( x ) I ν ( x ) ≤ x > ν ≥ − (see [13, Lemma3]). To see that L ν ( x ) < I ν ( x ), x > ν ≥ − , we recall that the modified Struvefunction of the second kind is defined by M ν ( x ) = L ν ( x ) − I ν ( x ). We can readilysee that M ν ( x ) <
0, for x > ν > − , from its integral representation (see [28,formula 11.5.4]), and M − ( x ) < x >
0, can be seen by using the formulas in(A.3).(iii) We will make use of the following inequality of [31] for a ratio of modifiedBessel functions of the second kind:(3.13) K ν ( x ) K ν − ( x ) < ν − + q ( ν − ) + x x < ν − x , x > , ν > . We now obtain inequality (3.4) by applying inequality (3.13) and the upper boundin (3.2): xK ν +3 ( x ) L ν ( x ) = K ν +3 ( x ) K ν +2 ( x ) · xK ν +2 ( x ) L ν ( x ) < (cid:18) ν + 5 x (cid:19) ν + 2) √ π Γ( ν + ) . N INTEGRAL INVOLVING THE MODIFIED STRUVE FUNCTION OF THE FIRST KIND 9 (iv) We note that the ratio Γ( ν +2)Γ( ν +3 / is an increasing function of ν on [ − , ] (see[20]). Therefore using the upper bound in (3.2) we obtain that, for − ≤ ν ≤ and x > xK ν +2 ( x ) L ν ( x ) < + 2) √ π Γ( + ) = 32 , where we used that Γ( ) = √ π . Thus, we have proved inequality (3.5). Inequalities(3.6) and (3.7) are obtained similarly (making use of the upper bound in (3.2) andinequality (3.4)), and we omit the details. (cid:3) Proof of Lemma 3.3.
Fix x ∗ > − β . We consider the function u ν,β ( x ) = M ν,β ( x ∗ )e − βx x ν L ν +1 ( x ) − Z x e − βt t ν L ν ( t ) d t, and prove inequality (3.8) by showing that u ν,β ( x ) > x ≥ x ∗ .We first prove that u ν,β ( x ∗ ) >
0. To this end, we consider the function v ν,β ( x ) = e βx x ν L ν +1 ( x ) Z x e − βt t ν L ν ( t ) d t, and it suffices to prove that v ν,β ( x ∗ ) < M ν,β ( x ∗ ). We note that ∂v ν,β ( x ) ∂β = e βx x ν L ν +1 ( x ) Z x ( x − t )e − βt t ν L ν ( t ) d t > , meaning that v ν,β ( x ) is an increasing function of β . Therefore, for 0 < β < v ν,β ( x ∗ ) < e x ∗ x ν ∗ L ν +1 ( x ∗ ) Z x ∗ e − t t ν L ν ( t ) d t < x ∗ ν + 1 (cid:18) L ν ( x ∗ ) L ν +1 ( x ∗ ) + 1 (cid:19) < x ∗ ν + 1 (cid:18) ν + 3 + x ∗ x ∗ + 1 (cid:19) = 2 ν + 3 + 2 x ∗ ν + 1 ≤ M ν,β ( x ∗ ) , where the second inequality is clear from the integral formula (1.3) and we appliedLemma 3.1 to obtain the third inequality.We now prove that u ′ ν,β ( x ) > x > x ∗ . A calculation using the differentiationformula (A.2) followed by an application of inequality (A.8) gives that u ′ ν,β ( x ) = M ν,β ( x ∗ ) dd x (cid:0) e − βx x − · x ν +1 L ν +1 ( x ) (cid:1) − e − βx x ν L ν ( x )= M ν,β ( x ∗ )e − βx x ν (cid:0) L ν ( x ) − x − L ν +1 ( x ) − β L ν +1 ( x ) (cid:1) − e − βx x ν L ν ( x ) > M ν,β ( x ∗ )e − βx x ν (cid:0) − β − x − ) L ν ( x ) − e − βx x ν L ν ( x ) ≥ (cid:18) − β − x − − β − x − ∗ − (cid:19) e − βx x ν L ν ( x ) > , for x > x ∗ . This completes the proof. (cid:3) Proof of Lemma 3.4. (i) We obtain inequality (3.10) by bounding the expressione βx K ν +3 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t for x ∈ (0 , x ∗ ) and x ∈ [ x ∗ , ∞ ), where x ∗ = C − β for some C > x ∈ (0 , x ∗ ). Observe that ∂∂β (cid:18) e βx K ν +3 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t (cid:19) = e βx K ν +3 ( x ) x ν − Z x ( x − t )e − βt t ν L ν ( t ) d t > . Since 0 < β <
1, we therefore have that, for x ∈ (0 , x ∗ ),e βx K ν +3 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t < e x K ν +3 ( x ) x ν − Z x e − t t ν L ν ( t ) d t< ν + 1 x K ν +3 ( x ) (cid:0) L ν ( x ) + L ν +1 ( x ) (cid:1) < ν + 1 (cid:18) x ∗ + 9 (cid:19) = 12 ν + 1 (cid:18) C − β ) (cid:19) < ν + 1)(1 − β ) (cid:18) C (cid:19) =: T , where we used (1.3) to bound the integral in the second step, and inequalities (3.6)and (3.7) to obtain the third inequality.Suppose now that x ∈ [ x ∗ , ∞ ). Let M ν,β ( x ∗ ) be defined as per (3.9). Boundingthe integral by inequality (3.8) gives thate βx K ν +3 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t < e βx K ν +3 ( x ) x ν − · M ν,β ( x ∗ )e − βx x ν L ν +1 ( x )= M ν,β ( x ∗ ) xK ν +3 ( x ) L ν +1 ( x ) < M ν,β ( x ∗ )= 158 max (cid:26) ν + 1 (cid:18) ν + 3 + 2 C − β (cid:19) , C ( C − − β ) (cid:27) ≤ max (cid:26) C )8(2 ν + 1)(1 − β ) , C C − ν + 1)(1 − β ) (cid:27) =: max { T , T } , where we used inequality (3.7) to obtain the second inequality and we used that − < ν ≤ to obtain the third inequality.It is readily checked that T ≥ T if C ≤
4. Equating T = T gives a quadraticequation for C with positive solution C = √ − = 1 . . . . . Thereforee βx K ν +3 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t < ν + 1)(1 − β ) (cid:18) · . (cid:19) = 13 . ν + 1)(1 − β ) < ν + 1)(1 − β ) . (ii) The proof of inequality (3.11) is similar to that of inequality (3.10). Let x ∗ = − β ) . By a similar argument, we have that, for x ∈ (0 , x ∗ ),e βx K ν +2 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t < ν + 1 x K ν +2 ( x ) (cid:0) L ν ( x ) + L ν +1 ( x ) (cid:1) < x ∗ ν + 1 (cid:18)
32 + 1 (cid:19) = 154(2 ν + 1)(1 − β ) , (3.14) N INTEGRAL INVOLVING THE MODIFIED STRUVE FUNCTION OF THE FIRST KIND11 where we applied inequalities (3.5) and (3.3) to get the second inequality. Supposenow that x ∈ [ x ∗ , ∞ ). Using inequality (3.8) gives thate βx K ν +2 ( x ) x ν − Z x e − βt t ν L ν ( t ) d t < M ν,β ( x ∗ ) xK ν +2 ( x ) L ν +1 ( x ) < M ν,β ( x ∗ )= max (cid:26) ν + 1 (cid:18) ν + 3 + 31 − β (cid:19) , − β (cid:27) ≤ max (cid:26) ν + 6(2 ν + 1)(1 − β ) , − β (cid:27) = 2 ν + 6(2 ν + 1)(1 − β ) < ν + 1)(1 − β ) , (3.15)where we used (3.3) to get the second inequality and we used that − < ν ≤ toobtain the third and fourth inequalities. We complete the proof by noting that thebound (3.15) is greater than the bound (3.14). (cid:3) Proofs of main results
Proof of Theorem 2.1. (i) Let x > − < ν ≤
0. Using integrationby parts and the differentiation formula (A.2) gives that Z x e − βt t ν L ν ( t ) d t = − β e − βx x ν L ν ( x ) + 1 β Z x e − βt t ν L ν − ( t ) d t, where we used that lim x ↓ x ν L ν ( x ) = 0, for ν > − (see A.4)). One can checkthat the integrals exist for ν > − by using the limiting form (A.4). By using theidentity (A.1) and rearranging we obtain that Z x e − βt t ν L ν +1 ( t ) d t + 2 ν Z x e − βt t ν − L ν ( t ) d t − β Z x e − βt t ν L ν ( t ) d t = e − βx x ν L ν ( x ) − Z x e − βt t ν √ π ν Γ( ν + ) d t. (4.1)Using inequality (A.8) to bound the first integral and making use of the assumptionthat ν ≤ Z x e − βt t ν L ν ( t ) d t > − β (cid:26) e − βx x ν L ν ( x ) − Z x e − βt t ν √ π ν Γ( ν + ) d t (cid:27) . Finally, we use a change of variable to evaluate the integral R x e − βt t ν d t = β ν +1 γ (2 ν +1 , βx ), which gives us inequality (2.1).(ii) Suppose now that ν ≥ . A rearrangement of (4.1) gives that Z x e − βt t ν L ν +1 ( t ) d t − β Z x e − βt t ν L ν ( t ) d t = e − βx x ν L ν ( x ) − ν Z x e − βt t ν − L ν ( t ) d t − Z x e − βt t ν √ π ν Γ( ν + ) d t = e − βx x ν L ν ( x ) − ν Z x e − βt t ν − L ν ( t ) d t − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) . (4.2) We use inequality (A.8) to bound the first integral on the left-hand side in (4.2),and then divide through by (1 − β ) and apply inequality (A.8) again to obtain Z x e − βt t ν L ν ( t ) d t > − β (cid:26) e − βx x ν L ν ( x ) − ν Z x e − βt t ν − L ν ( t ) d t − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) > − β (cid:26) e − βx x ν L ν ( x ) − ν Z x e − βt t ν − L ν − ( t ) d t − γ (2 ν + 1 , βx ) √ π ν β ν +1 Γ( ν + ) (cid:27) . (4.3)Lastly, we bound the integral R x e − βt t ν − L ν − ( t ) d t using inequality (2.10) (whichcan be done because ν ≥ ), which gives us inequality (2.2).(iii) Let ν > −
1, which ensures that all integrals in this proof of inequality (2.3)exist. We start with the same integration by parts to part (i), but with ν replacedby ν + 1: Z x e − βt t ν +1 L ν +1 ( t ) d t = − β e − βx x ν +1 L ν +1 ( x ) + 1 β Z x e − βt t ν +1 L ν ( t ) d t, (4.4)where it should be noted that we used that lim x ↓ x ν +1 L ν +1 ( x ) = 0 for ν > − R x e − βt t ν +1 L ν ( t ) d t < x R x e − βt t ν L ν ( t ) d t , x >
0, which holds because L ν ( t ) > t > ν > −
1. Applying this inequalityto (4.4) and rearranging gives(4.5) Z x e − βt t ν L ν ( t ) d t > e − βx x ν L ν +1 ( x ) + βx Z x e − βt t ν +1 L ν +1 ( t ) d t. We can use (4.5) to obtain another inequality Z x e − βt t ν L ν ( t ) d t> e − βx x ν L ν +1 ( x ) + βx (cid:18) e − βx x ν +1 L ν +2 ( x ) + βx Z x e − βt t ν +2 L ν +2 ( t ) d t (cid:19) = e − βx x ν L ν +1 ( x ) + β e − βx x ν L ν +2 ( x ) + β x Z x e − βt t ν +2 L ν +2 ( t ) d t, and iterating gives inequality (2.3). In performing this iteration, it should be notedthat the series P ∞ k =0 β k L ν + k +1 ( x ) is convergent. This can be seen by applyinginequality (A.8) (since ν > −
1) to obtain that, for all x > P ∞ k =0 β k L ν + k +1 ( x ) < L ν +1 ( x ) P ∞ k =0 β k = L ν +1 ( x )1 − β , with the assumption that 0 < β < x → ∞ .To this end, we note the following limiting forms, which hold for all ν > − < β < Z x e − βt t ν L ν ( t ) d t ∼ √ π (1 − β ) x ν − / e (1 − β ) x , x → ∞ , (4.6) e − βx x ν L ν + n ( x ) ∼ √ π x ν − / e (1 − β ) x , x → ∞ , n ∈ R , (4.7) N INTEGRAL INVOLVING THE MODIFIED STRUVE FUNCTION OF THE FIRST KIND13 where (4.7) is immediate from (A.5), and (4.6) follows from using (A.5) and astandard asymptotic analysis. The tightness of inequalities (2.1) and (2.2) in thelimit x → ∞ follows immediately from (4.6) and (4.7). To show that inequality(2.3) is tight as x → ∞ we just need to additionally use that P ∞ k =0 β k = − β , since0 < β < (cid:3) Proof of Theorem 2.2. (i) Rearranging inequality (3.10) gives that, for x > − < ν < , 0 < β < Z x e − βt t ν L ν ( t ) d t < ν + 1)(1 − β ) e − βx x ν − K ν +3 ( x ) . From the bound K ν +3 ( x ) < x L ν +1 ( x ), which is a rearrangement of the lower boundin (3.2), we obtain that, for x > − < ν < , 0 < β < Z x e − βt t ν L ν ( t ) d t < ν + 1)(1 − β ) e − βx x ν L ν +1 ( x ) . Using that 2 ν + 1 > − < ν < gives us inequality (2.4) for the case − < ν < . Inequality (2.4) can in fact be seen to hold for all ν > − , by notingthat the upper bound in inequality (2.4) is strictly greater than the the upper boundin inequality (2.10) (due to [15]), which is valid for ν ≥ .(ii) We argue as in part (i), but we apply inequality (3.11), rather than inequality(3.10), and then use the bound K ν +2 ( x ) < x L ν ( x ).(iii) The proof proceeds exactly as that of inequality (2.2), with the sole modifica-tion being that we use (2.4) to bound the integral on the right-hand side of (4.3),instead of inequality (2.10). The tightness of inequality (2.6) in the limit x → ∞ isestablished by the same argument as that used in part (iv) of the proof of Theorem2.1. (cid:3) Proof of Proposition 2.3. (i) To get inequality (2.7), in part (i) of the proof ofTheorem 2.1 use inequality (A.8) to bound the third integral in (4.1), instead ofthe first integral.(ii) To get inequality (2.8), in part (ii) of the proof of Theorem 2.1 use (A.8) tobound the second integral in (4.2), instead of the first integral.(iii) By studying the proof of inequality (2.6), it can be seen that the above alter-ation that gave us inequality (2.8) instead of inequality (2.2) can also be used togive us inequality (2.9). (cid:3)
Appendix A. Basic properties of modified Struve and modified Besselfunctions
In this appendix, we present some basic properties of the modified Struve func-tion of the first kind L ν ( x ) and the modified Bessel functions I ν ( x ) and K ν ( x ) thatare used in this paper. All formulas are given in [28], except for the inequalitywhich was obtained by [5].The modified Struve function L ν ( x ) is a regular function of x ∈ R , and is positivefor all ν ≥ − and x >
0. The modified Bessel functions I ν ( x ) and K ν ( x ) are alsoboth regular functions of x ∈ R . For x >
0, the functions I ν ( x ) and K ν ( x ) arepositive for ν ≥ − ν ∈ R , respectively. The modified Struve function L ν ( x ) satisfies the following recurrence relation and differentiation formula L ν − ( x ) − L ν +1 ( x ) = 2 νx L ν ( x ) + ( x ) ν √ π Γ( ν + ) , (A.1) dd x (cid:0) x ν L ν ( x ) (cid:1) = x ν L ν − ( x ) . (A.2)We have the following special cases(A.3) L − ( x ) = r πx sinh( x ) , I − ( x ) = r πx cosh( x ) . We also have the following asymptotic properties: L ν ( x ) ∼ x ν +1 √ π ν Γ( ν + ) (cid:18) x ν + 3) (cid:19) , x ↓ , ν > − , (A.4) L ν ( x ) ∼ e x √ πx (cid:18) − ν − x (cid:19) , x → ∞ , ν ∈ R , (A.5) K ν ( x ) ∼ ν − Γ( ν ) x ν , x ↓ , ν > , (A.6) K ν ( x ) ∼ r π x e − x (cid:18) ν − x (cid:19) , x → ∞ , ν ∈ R . (A.7)It was shown by [5] that, for x > L ν ( x ) < L ν − ( x ) , ν ≥ . Other inequalities for the modified Struve function L ν ( x ) are given in [5, 6, 16, 23],some of which improve on inequality (A.8). Acknowledgements.
I would like to thank the referees for their helpful commentsand suggestions that helped me improve the presentation of my paper.
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