Functional Inequalities and Bounds for the Generalized Marcum Function of the Second Kind

In this paper, we consider the generalized Marcum function of the second kind as an analogous function of the so-called generalized Marcum Q-function. We provide the log-convexity (log-concavity) property for its unit complement and improve some of our previous results on it. One of the transformed functions of the generalized Marcum function of the second kind is discussed in details in this paper. This form of the generalized Marcum function of the second kind supplies various important inequalities. We also discuss the Turán type inequality for the generalized Marcum Q-function. Additionally, we provide the bounds for the generalized Marcum function of the second kind as well as for its symmetric difference.

This function is an analogous function of the so-called generalized Marcum Q-function, which has applications in radar communication, see for example [13,14]. In these articles [13,14] the authors have used the following equatioñ as a definition of the generalized Marcum Q-function, while many others have used the following definition The generalized Marcum Q-function Q ν (a, b) and its particular case Q 1 (a, b), the so-called Marcum function, are important functions of the electrical engineering literature, and their analytical properties, approximations and computations have been investigated by many researchers (mostly engineers) in the last decades, see for example [21] and the references therein. It turned out that the generalized Marcum Q-function possesses some quite interesting properties and its investigation is rather interesting from the mathematical point of view too. From the above two definitions we can conclude that and analogously to Q ν ( √ a, √ b), we can derive by replacing a and b by √ a and √ b respectively, in (1.1). In view of [18, eq. 10.27.3] K −ν (x) = K ν (x) and [18, eq. 8.6.6] Both forms (1.1) and (1.2) have been used in [4], which contains a detailed study of the generalized Marcum function of the second kind. In particular, the study includes monotonicity, convexity, recurrence relation, closed form expression, and tight bounds for the generalized Marcum function of the second kind. In [5], extremely tight bounds of the generalized Marcum function of the second kind were obtained. It is interesting to note that the generalized Marcum function of the second kind is the survival function of the truncated distribution of a special case of the modified Bessel distribution of the second kind considered by Nadarajah [16], see [4] for more details. Therefore, in view of [4,Eq. (1.4)] the corresponding cumulative distribution function (the unit complement to the generalized Marcum function of the second kind) is given by The above function is similar to the unit complement to the generalized Marcum Q-function, which has been discussed in [21]. Moreover, the authors of [21] have found some kind of analytical properties for this function. We also have the transformed form R ν ( √ a, √ b) of the generalized Marcum function of the second kind in (1.2), which has a similar form as the normalized function f a (x) [2], defined by Alzer and Baricz [2] have derived various inequalities for the normalized function f a (x). Motivated by [2,21], in this paper, our aim is to present various inequalities involving the transformed form of the generalized Marcum function of the second kind with some analytical properties for S ν (a, b). We have also found some Turán type inequalities for the generalized Marcum function of the second kind and the generalized Marcum Q-function by a unified approach. However, Baricz and Sun have already developed the Turán type inequality for the generalized Marcum Q-function in [20], that is later on directly concluded by the log-concavity property of the function ν → Q ν (a, b) on [1, ∞] for all a, b ≥ 0, [21]. Additionally, we provide some bounds for the generalized Marcum function of the second kind and its symmetric difference, which is defined by analogously to the symmetric difference of the generalized Marcum Q-function discussed in [6,11]. All the main results that include the inequalities and analytical properties with some other results are provided in the next section, while Sect. 3 contains the proofs of the main results.

Main Results
This section is divided into five subsections: in the first subsection, we present inequalities for the transformed form (1.2) of the generalized Marcum function of the second kind. The second subsection contains several properties of the generalized Marcum function of the second kind (1.1) and its unit complement (1.3). This subsection also deals with the Turán type inequality for R ν (a, b), while the third subsection provides the Turán type inequality for the generalized Marcum Q-function. The fourth subsection is dedicated to some new bounds for the generalized Marcum function of the second kind, especially, to a uniform upper and lower bound. In addition, we also provide some lower and upper bounds for the symmetric difference of the generalized Marcum function of the second kind in the last subsection.

Inequalities for the Generalized Marcum Function of the Second Kind
In this subsection, we discuss the transformed generalized Marcum function of the second kind which is defined in Eq. (1.2). This transformed function is very important as we can see in [4], and it is used to obtain the monotonicity of the generalized Marcum function of the second kind with respect to the parameters a and ν. Now, we use this transformed function R ν ( √ a, √ b) to derive various inequalities. We use the notation R ν (a, b) for this transformed , for the sake of convenience, however it is used only in Theorem 1 and Lemma 2. The following theorem includes various inequality for R ν (a, b) which are motivated by the results of [2]. Before stating the theorem, we describe here a few notations: The power sum and power mean of order t are denoted by The other notations are and M 0 (x 1 , x 2 , . . . , x n ) = (x 1 , . . . , x n ) 1/n . Theorem 1. Let a > 0, ν > 0 and x 1 , . . . , x n , n ≥ 2, be positive real numbers.
a. If p ≥ 1 then the inequality holds. Its converse is also true if ν ≥ 1.
holds if and only if r ≥ 1.

d. The inequality
Rν (a,x1) + · · · + 1 Rν (a,xn) (2.4) holds if and only if s = ∞. e. The inequality ) holds for all a > 1 and bounds are the sharpest. f. Let p = 0 and q = 0 be real numbers. The inequality holds for all positive real numbers x and y, i.e., the function x → R ν (a, x p )) q is strictly subadditive on (0, ∞) if and only if pq > 0. g. The inequality holds for all positive real numbers x, y and z. h. The function x → R ν (a, x) is completely monotonic on (0, ∞) for all a > 0 and ν > 0.

Monotonicity Patterns of the Generalized Marcum Function of the Second Kind
From [4] we recall that the function b → R ν (a, b) is logarithmically concave on (0, ∞) for each ν > 3/2 and a > 0, and thus for all a > 0, ν > 3/2, b 1 , b 2 ≥ 0 and b 1 = b 2 , it satisfies the following inequalities: The left-hand side inequality is nothing but the new is better than used inequality, which has several applications in the economic theory, while the other part of the inequality is the inequality directly obtained by log-concave property of the function b → R ν (a, b). In part c of [4,Theorem 1] we also showed that the transformed generalized Marcum function of the second kind stated 35 Page 6 of 28Á. Baricz et al.

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in (1.2) is log-convex with respect to b on the interval (0, ∞) when a > 0 and ν ≥ 1. This property implies the inequality The right-hand side inequality is reversed to the new is better than used inequality, that sometimes called the new is worst then used (nws) inequality. In this subsection, we discuss similar kind of properties for the unit complement to the generalized Marcum function of the second kind which is defined in (1.3). In particular, we prove that the function b → S ν (a, b) is log-concave on (0, ∞) for all a > 0 and ν > 3/2. Unfortunately, the unit Our first theorem answers the question about logarithmic and geometric convexity/concavity of the unit complement to the generalized Marcum function of the second kind. Theorem 2. The following assertions are true: Part f of Theorem 2 gives the following inequality We also have the inequality due to the property of the function a → R ν (a, b)/c a,ν provided in part g of Theorem 2. In view of (1.3), the monotone property of the function S ν (a, b) can be obtained form the monotonicity of R ν (a, b), [4,Theorem 1]. Thus the function S ν (a, b) is monotonic increasing with respect to the parameter a and b but decreasing with respect to the parameter ν.

Remark 1. Part d and e of Theorem 2 provide similar inequalities for
ν such as we have (2.9) and (2.10) respectively for R ν (a, b). We also get the following inequalities which are based on log-concavity for all ν > 0 and a > 0 respectively. The right hand side of the inequality (2.8) is weaker than the inequality (2.12) since the function and ν > 0, while the log-concavity property gives stronger inequality than the geometric-concavity property for R ν (a, b). It is worth mentioning here that the inequality (2.8) is valid only for ν > 3/2 but by using the part c, we extend the range of ν from ν > 3/2 to ν > 0 for the inequality (2.8) since an increasing geometrically concave function is a log-concave function too.
for all a > 0 and ν ≥ 1, and this can be improved as follows.
. By using the same technique as we used in part e, the function for all a > 0 and ν ≤ 3/2. Combining this with part c of [4, Theorem 1] we get the following statement: for all a > 0 and ν ≥ 0.
Next, our aim is to find some Turán-type inequality for R ν (a, b). One way of getting such type of inequality is to find the log-concavity/convexity property for the functions with respect to their parameters. Our numerical results suggest the following conjectures: Remark 3. If Conjecture 1 is true then in view of the inequality (2.9) we get .
Taking into consideration the asymptotic formula (2.23) we have and it is easy to calculate Thus, the inequality (2.13) with the above discussion gives the following tight bounds for φ ν (a, b) Our numerical results also suggest the following conjecture. is monotonic increasing on the interval (0, ∞) for all a > 0 and ν > 0. Conjecture 3 is also supported by Remark 3 and if this conjecture is true then the inequality (2.13) is the sharpest one and thus cannot be further improved.

Turán Type Inequality of the Marcum Q-Function
It is well-known that the generalized Marcum Q-function has the log-concavity property with respect to all the parameters. This is proved in the papers [20,21]. With the help of log-concavity property of Q ν (a, b) we can have the Turán type inequality for the generalized Marcum Q-function. In this subsection, our aim is to find the Turán type inequality for the generalized Marcum Qfunction with another method. The method is same as we used in the previous subsection to obtain the Turán type inequality for the generalized Marcum function of the second kind. Now, we recall first some important formulae related to the generalized Marcum Q-function.
where a ≥ 0, b, ν > 0 and I ν stands for the modified Bessel function of the first kind. Integrating by parts on (2.14) yields Now, we give the main theorem of this subsection. From Theorem 4 we conclude the following corollary. ) is an increasing function on (0, ∞) for all ν > 1 and b > 0. Moreover, the Turán type inequality Proof. Re-write the recurrence relation (2.15) as This implies that the functions Q ν−1 (a, b)/Q ν (a, b) and Q ν (a, b)/Q ν (a, b) have the same monotonic property with respect to ν as well as a. Hence the result follows. Furthermore, the monotonicity of the function ν → Q ν (a, b)/Q ν+1 (a, b) gives the desired inequality.

Some New Bounds for the Generalized Marcum Function of the Second Kind
In [4,5], we have found various upper and lower bounds for generalized Marcum function of the second kind. These bounds were obtained by using the monotonicity of functions which involve the modified Bessel function of the second kind. In [4, Theorem 8], we found a uniform upper bound for the generalized Marcum function of the second kind, which is given by we conclude the following bounds for the Marcum function of the second kind π 2 c a,1 dt.
(2.17) In view of part g of [4, Theorem 1] the function ν → R ν (a, b) is increasing on (0, ∞) for all a > 0 and b ≥ 0 and a well-known result is that a → Q ν (a, b) is increasing on (0, ∞), see [21,Theorem 1]. Thus, the inequalities (2.16) and (2.17) give the following (2.18) Next, by using the monotonicity of the function t → e − t 2 2 and the differential formula [18, 10.29 in Eq. (1.1), we get the following upper bound , where ν > 1.
(2.21) Then we have By using Similarly, we can find that Hence we have the following uniform upper and lower bound we can conclude that the bounds obtained in (2.22) are tight when b → ∞ as well as b → 0.

Bounds for the Symmetric Difference of the Generalized Marcum Function of the Second Kind
In this subsection, we discuss the symmetric difference of generalized Marcum function of the second kind. This work is motivated by the results on the symmetric difference of the generalized Marcum Q-function. The symmetric difference of the generalized Marcum Q-function is defined as

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and this function has been studied by various authors, see [6,11] and references therein, due to its use in digital communication. Since the function Q ν (a, b) has opposite monotonicity property with respect to the parameter a and b, the function a → ΔQ ν (a, b) is increasing on (0, ∞), while the function b → ΔQ ν (a, b) is decreasing on (0, ∞). In a similar manner, we define the symmetric difference of the generalized Marcum function of the second kind Unfortunately, the study of the symmetric difference of generalized Marcum function of the second kind is more complicated than the study of symmetric difference of Marcum Q-function because they do not have the same monotonic behavior throughout the interval (0, ∞). All the same, it is easy to find bounds for the function ΔR ν (a, b), which gives a certain idea about its value. We use the bounds of the generalized Marcum function of the second kind obtained in [4,5] (2.29) [4, Eq. 2.11] If ν ≥ 3/2 and b ≥ a > 0, then (2.30) (2.31) [4, Eq. 2.18] If ν ≥ 3/2 and a > b > 0, then From the above mentioned bounds, we can find the bounds for the symmetric difference of R ν (a, b). It is evident from the definition of symmetric difference that ΔR ν (a, b) = −ΔR ν (b, a). Therefore, we only supply bounds for the case a > b > 0, while the other case b > a > 0 will follow by the above relation. It is clear that for ν ≥ 1, the upper bound for ΔR ν (a, b) is obtained by subtracting (2.29) from (2.25) Another upper bound is as follows

Proof of Main Results
Here, first we give some important lemmas which will be useful to prove our main results.
If F is concave on [0, ∞), then the reversed inequality holds.
The following lemma is a useful tool to achieve the inequalities which are given in Theorem 1.

Lemma 2.
The following are true: Proof. a. This follows from part c of [4, Theorem 1] and Remark 2. , 0), the property with respect to b of R ν (a, b) can be verified from the function A ν (a, b). By some easy calculation we have It is clear that (3.1) Further differentiation gives In view of the result [9, p. 583] for t > 0 and ν ≥ 0, , J ν and Y ν stand for the Bessel function of the first and second kind, respectively, Eq. (3.2) leads to For the case when ν < 1, Eq. (3.1) reduces to .

In view of Lemma 3 the above equation gives
when a > 1 and ν < 3/2. Hence the function u → log(u ν−1 ) is log-convex on (0, ∞) when a ≥ 1 and ν > 0. Since the log-convexity property remains invariant after integration, A ν (a, b 1/a ) is log-convex for all a ≥ 1 and ν > 0.
b. It is well-known that the power mean t → M t (x 1 , . . . , x n ) is an increasing function on (0, ∞), see [8, p. 26]. This implies that the function t → R ν (a, M t (x 1 , . . . , x n )) is decreasing on (0, ∞). Thus we have for all r ≥ 1. Now in view of above inequality, it is enough to prove the left hand inequality of (2.2) only for r = 1. Using part b of Lemma 2 and the fact that every log-convex function is also a convex function, we have R ν (a, M 1 (x 1 , . . . , x n )) = R ν a, x 1 + · · · + x n n ≤ R ν (a, x 1 ) + · · · + R ν (a, x n ) n . Since the function x → R ν (a, x) is a decreasing function on (0, ∞), for all i = 1, 2, . . . , n R ν (a, x i ) ≤ R ν (a, M s (x 1 , . . . , x n )) when s = −∞.
By adding these inequalities, we get the right-hand side of the inequality (2.2). Assume that the left-hand side of (2.2) is true for any positive real numbers x 1 , x 2 , . . . , x n . Then for x, y > 0, we have M r (x, y, . . . , y) It is evident that V a,r (y, y) = 0 and ∂ ∂x V a,r (x, y) This implies In view of (2.24) and [18, 10.31.3] Now take into consideration a well-known result from real analysis (that is, if a + > 0 for all > 0, then a ≥ 0), and then Eqs. (3.4) and (3.5) together give r ≥ 1 for all ν > 2. This leads us to r ≥ 1. Now for the case 0 < ν < 1, we rewrite the Eq. (3.4) By using a similar argument as we used above, we get r ≥ 1 since lim y→0 ay 2 Suppose that there exists a real number s such that the right-hand side of (2.2) is satisfied for all x 1 , . . . , x n > 0. Consider s ≥ 0. If x 1 → ∞, then clearly right hand inequality of (2.2) does not hold. Consider s < 0. Let us take x 1 = x, x 2 = · · · = x n = y and allowing y tends to ∞, then for all x > 0 By using the fact that the function t → K ν (t) is decreasing on (0, ∞), we have (1−c) ).
As we know c = n − 1 s > 1, therefore for all x > x * we have where x * is the unique root of the equation This shows that φ ν (a, x) is an increasing function on (x * , ∞) and hence which is a contradiction. c. From part a of Lemma 2 and the fact every log-convex function is convex, we have Since the power mean is an increasing function with respect to its order [8, p. 26], Conversely, suppose that the inequality (2.3) is true for all x 1 , . . . , x n . Then we have (A ν (a, M r (x 1 , . . . , x n ))) n ≤ A ν (a, x 1 ) . . . A ν (a, x n ).
Put x 1 = x, x 2 = · · · = x n = y, then the above inequality can be written as where, M r denotes M r (x, y, . . . , y). In view of asymptotic formula (2.23) and r < 1 we have The above two limits with inequality (3.7) give a contradiction. Hence r ≥ 1.
This gives us Now, as the function t → 1/R ν (a, t) is an increasing function on (0, ∞), we get . . . , x n ) . ≤ (R ν (a, x 1 ) · · · R ν (a, x n )) 1/n , and the inequality (2.4) together imply Therefore by part c we have s ≥ 1. Now set x 1 = x, x 2 = · · · = x n = y and allowing y tends to 0, then the inequality (2.4) implies Taking into consideration the asymptotic formula (2.23) of the modified Bessel function of the second kind, we have Since c = n −1/s < 1, then (3.9) gives which is a contradiction for (3.8). Therefore, s is not a real number. Hence s = ∞. e. Part b of Lemmas 2 and 1 imply the inequality (2.5). The bounds are sharpest since if we take all x i tend to ∞ then we get value zero and if we set x 1 = x, x 2 = · · · = x n = y and taking y tends to zero then we get the value n − 1.
This gives the required result. On differentiating R ν (a, x) with respect to x, we have The Eq. (3.12) with the above relation show that −R ν (a, x) is the product of two completely monotonic functions. Hence x → R ν (a, x) is completely monotonic on (0, ∞).
Now, we continue with the proof of Theorem 2.
Proof of Theorem 2. a. By part a of [4, Lemma 2], the function t → t ν−1 K ν−1 (at) is log-concave on (0, ∞) for all a > 0 and ν > 3/2. It is quite clear that the function t → t ν e − t 2 2 is a log-concave function on (0, ∞). Thus the product of these two functions, i.e., the function t → t ν e − t 2 2 K ν−1 (at) is also a log-concave function on (0, ∞) for a > 0 and ν > 3/2 as the product of two log-concave functions is again a log-concave function. Therefore, the probability density function t → c a,ν a ν−1 t ν e − t 2 +a 2 2 K ν−1 (at) is log-concave on (0, ∞) for all ν > 3/2. Thus, in view of [3, Theorem 1] the cumulative distributive function of a log-concave probability density function is log-concave, the cumulative distributive function i.e., S ν (a, b) is log-concave on (0, ∞) for all ν > 3/2 and a > 0. Furthermore, the inequality (2.8) follows by Lemma 1 and the log-concave property of the function b → S ν (a, b). b. On replacing a and b by √ a and √ b respectively, in the Eq. (1.3), we have Formula (2.19) gives and it is well known that the function t → e − t 2 is decreasing on (0, ∞). Thus, the function is decreasing on (0, ∞) for all ν > 0 and a > 0. Thus, we conclude that the function b → S ν ( √ a, √ b) is concave on (0, ∞) and hence log-concave. By replacing a by a 2 we get the required result.