Inequalities on an extended Bessel function

This paper studies an extended Bessel function of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {}_{a}\mathtt{B}_{b, p, c}(x):= \sum _{k=0}^{\infty }\frac{(-c)^{k}}{k! \Gamma { ( a k +p+\frac{b+1}{2} ) } } \biggl( \frac{x}{2} \biggr) ^{2k+p}. $$\end{document}Bb,p,ca(x):=∑k=0∞(−c)kk!Γ(ak+p+b+12)(x2)2k+p. Representation formulations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{a}\mathtt{B}_{b,p, c}$\end{document}Bb,p,ca are derived in terms of the parameters a, b, and p. An important consequence is the derivation of an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(a+1)$\end{document}(a+1)-order differential equation satisfied by the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{a}\mathtt{B}_{b,p, c}$\end{document}Bb,p,ca. Interesting functional inequalities are established, particularly for the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a=2$\end{document}a=2, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c=\pm \alpha^{2}$\end{document}c=±α2. Monotonicity properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{a}\mathtt{B}_{b,p, c}$\end{document}Bb,p,ca are also studied for non-positive c. Log-concavity and log-convexity properties in terms of the parameters d and p are respectively investigated for the closely related function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {}_{a}\mathcal{B}^{d}_{b,p, c}(x): =\sum _{k=0}^{\infty }\frac{(-c/4)^{k} \Gamma { ( p+\frac{b+1}{2} ) }}{ \Gamma{ ( k+1 ) } \Gamma { ( ak+p+\frac{b+1}{2} ) }} \frac{(d)_{k}}{k!}x^{k}, $$\end{document}Bb,p,cda(x):=∑k=0∞(−c/4)kΓ(p+b+12)Γ(k+1)Γ(ak+p+b+12)(d)kk!xk, which leads to direct and reverse Turán-type inequalities.


Introduction
The Bessel function of the first kind of order p given by is a particular solution of the homogeneous Bessel differential equation x 2 y (x) + xy (x) + x 2p 2 y(x) = 0.
Here denotes the gamma function. A solution of the homogeneous modified Bessel equation x 2 y (x) + xy (x)x 2 + p 2 y(x) = 0 gives the modified Bessel function of order p Because of its importance, the Bessel function and other special functions are of continued interest to the wider scientific community. The Bessel function and its variations have gone through several generalizations, see, for example, [1][2][3][4][5][6]. These generalized functions have also been framed as complex-valued analytic functions in the unit disk. Geometric properties of such functions have been studied, notably in [7][8][9][10][11][12][13].
Among the several generalized forms, perhaps a more complete generalization is that given by Baricz in [1]. In this case, the generalized Bessel function takes the form 2 ) x 2 2k+p (1) for a ∈ N = {1, 2, 3, . . .}, and b, p, c, x ∈ R. It is evident that the function a B b,p,c converges absolutely at each x ∈ R. Earlier, Galué [14] introduced a generalization of the Bessel function of the form a J p (x) := ∞ k=0 (-1) k k! (ak + p + 1) x 2 2k+p , x ∈ R, a ∈ N.
Apparently not much has been investigated for the extended Bessel function given by (1). Presumably such extensions would readily follow from recent results along similar used arguments, albeit involving intense computations. Still several pertinent questions remain, which include the question on how the parameter a influences the shape of the differential equation satisfied by a B b,p,c . It is the aim of this paper to complement and to fill the void of earlier investigations on the Bessel function and its extensions.
The connection between the parameters a, b, and p in the representation formulae and recurrence relation for a B b,p,c are derived in Section 2. An important consequence is the derivation of an (a + 1)-order differential equation satisfied by the function a B b,p,c . As applications, new functional inequalities for a B b,p,-α 2 are obtained, particularly in the case a = 2.
Section 3 is devoted to the investigation of the monotonicity properties of a B b,p,c for non-positive c, as well as for the normalized generalized Bessel function. Log-concavity and log-convexity properties in terms of the parameters d and p are also respectively investigated for the closely related function As a consequence, direct and reverse Turán-type inequalities [15] are obtained.

General representation formulations and applications
This section aims to find representation formulations, including integral representations, for the generalized function a B b,p,c in terms of the parameters a, b, and p. A starting point is the Gauss multiplication theorem [16] for the gamma function, which states that l = -ak, -ak -1, -ak -2, . . . , and k ∈ N. Here (α) k is the Pochhammer symbol defined by Choosing l = p + (b + 1)/2, it is evident from (1) that which leads to the following representation in terms of the generalized hypergeometric function (see [17]): Proposition 1 Let a ∈ N, and b, p, c, x ∈ R. Then Another representation formula can be expressed in terms of the order a = 1. In the sequel, we shall simply write B b,p,c := 1 B b,p,c . Thus Proposition 2 Let a ∈ N, and b, p, c, x ∈ R. Then Proof It is clear from (2) that Thus, x a a/2 . Remark 1 As a first application, let a = 2. In this case, Proposition 2 yields For obtaining recurrence relations, first differentiate (1) to yield Expanding the left side of the above equation yields Yet another form for x a B b,p,c is obtained from Expanding the left side of the above relation, it follows that Thus (5) and (6) lead to the following recurrence relation.

Proposition 3
Let a ∈ N, and b, p, c, x ∈ R. Then We next find an (a + 1)-order differential equation satisfied by a B b,p,c from the recurrence relations (5) and (6) (see also [18]). Theorem 1 Let the operator D be given by D := x(d/dx). For each k = 1, . . . , a, the generalized Bessel function a B b,p,c satisfies the differential equation In particular, the generalized Bessel function a B b,p,c is a solution of the differential equation Proof The proof is by induction. In terms of the differential operator D, identity (6) takes the form Now identity (5) gives Applying the operator D to both sides of (9), the latter equation leads to This establishes (7) for k = 1. Assume now that (7) holds for k = n. It follows from (6) that Applying the operator D to both sides of (7) for k = n, the above equation shows that The induction formula allows us to rewrite the final term above in the form that is, Remark 2 For a = 1, the differential equation (8) reduces to This is the differential equation considered by Baricz [2] in his study on the unification of Bessel, modified Bessel, spherical Bessel, and modified spherical Bessel functions. Thus the differential equation yields the Bessel function of the first kind of order p when b = c = 1, and the modified Bessel function of the first kind of order p when b = 1 and c = -1. In the case b = 2 and c = 1, there results the spherical Bessel function of order p. For a = 2, (8) reduces to Thus its particular solution is 2 B b,p,c , which from Proposition 1 can be expressed in the We conclude this section by establishing two integral representations for a B b,p,c . For this purpose, first recall the integral form of the beta function B(x, y) [16,17] given by for Re x > 0, Re y > 0. Replacing x by (ak + 1) and y by (2p + b -1)/2 in (10), we get Then the generalized Bessel function a B b,p,c takes the form which establishes the following proposition.

Proposition 4
Let a ∈ N, and b, p, c, x ∈ R. Then Remark 3 The particular cases a = b = 1 = ±c in Proposition 4 respectively lead to Another integral representation is the following.
Proof Replace x by (k + 1/2) and y by ( where p > (a -1b)/2. On the other hand, (4) yields Thus (11) and (12) shows that the generalized Bessel function a B b,p,c takes the form Now the Legendre duplication formula (see [16,17]) shows that This reduces (13) to the desired representation and completes the proof.
Remark 4 For another application, choose c = ±α 2 . Then Proposition 5 leads to cos αxt a a/2 dt, Substituting t = cos θ yields The particular case a = b = 1 = α in the above representations gives respectively the integral representation for the classical Bessel and modified Bessel functions of order p: These integrations for J p and I p can also be found in [16, 9.

Monotonicity and consequences
Investigations into the monotonicity properties of the generalized function a B b,p,c hinges on the following result of Biernacki and Krzyż [19].

Lemma 1 ([19])
Suppose f (x) = ∞ k=0 a k x k and g(x) = ∞ k=0 b k x k , where a k ∈ R and b k > 0 for all k. Further suppose that both series converge on |x| < r. If the sequence {a k /b k } k≥0 is increasing (or decreasing), then the function x → f (x)/g(x) is also increasing (or decreasing) on (0, r).
Evidently, the above lemma also holds true when both f and g are even functions, or both odd.
Proof (a) From (1) it is evident that and α k, .
Write w k = α k,p,a /α k,q,d ; since d ≥ a and q ≥ p, it follows that The result now readily follows from Lemma 1.
(b) Let q ≥ p > -(b + 1)/2. It follows from part (a) that d dx It now follows from (5) that (c) Let β k,p,a := (2k + p)α k,p,a . Then the quotient x a B b,p,c / a B b,p,c can be written as Clearly, the sequence {β k,p,a /α k,p,a } k≥0 = {2k + p} k≥0 is increasing, and hence Lemma 1 shows that the function x → x a B b,p,c / a B b,p,c is increasing on (0, ∞).

Next consider the normalized function
Also let 1 1 be the confluent hypergeometric function The next result discusses the monotonicity property of rational functions involving a B b,p,c .
Another function of interest is that given by Note ,p,c is given by (14). The following result by Karp and Sitnik [20,Theorem 1] is required to deduce the log-concavity of a B d b,p,c in terms of the parameter d.
Let be the digamma function given by (p) = (p)/ (p). Then evidently Note that [16, p. 260] has the explicit form This implies that for all k ∈ {0, 1, 2, . . .} and p > - which holds whenever (17) for all i, j ∈ N. Let Then Similarly, Next, for i ≥ j, relations (18) and (19) show that inequality (17) is equivalent to Since q ≥ p, this can be further simplified to showing The latter inequality clearly holds true whenever λ 1 ≥ λ 2 . To see that this is indeed the case, for q ≥ p, let . Since x → (ax + y) is log-convex, it follows that φ (x) ≥ 0. Thus φ(i) ≥ φ(j) for i ≥ j, and consequently λ 1 ≥ λ 2 . This validates inequality (16).
We shall show that the sequence b k = {f k /f k-1 } is decreasing. A calculation gives , and so we need to show that the function ξ : (0, ∞) → R given by Since the digamma function is known to be increasing on (0, ∞) for p > (2ab -1)/2 and x > 0, it follows that Thus ξ is indeed decreasing, and Lemma 2 shows that the function The results of parts (a) and (b) in Theorem 4 in the case d = 1 were also obtained by Baricz [1,Theorem 3,Theorem 4].
Remark 5 Theorem 4 has interesting consequences, among which is the Turán-type inequality for the function a B d b,p,c given by (15). From the definition of log-convexity, it follows from Theorem 4(a) that where α ∈ [0, 1], p 1 , p 2 > -(b + 1)/2, and x > 0. Choosing α = 1/2, p 1 = pν and p 2 = p + ν, the above inequality yields On the other hand, the log-concavity of d → a B d b,p,c implies that Thus (20) and (21) lead to direct and reverse Turán-type inequalities Remark 6 For d = 2, it follows from (15) that where a B b,p,c is given by (14). With d = 1 and μ = 1 in (21), . Thus (22) shows that Remark 7 Inequality (16) leads to a generalization of the Turán-type inequality Inequality (16) also yields . The next result gives a dominant function for a B b,p,-α 2 .
Theorem 5 Let p ≥ -(b + 1)/2 and x ≥ 0. Then Proof Clearly the estimate trivially holds for x = 0. Let x > 0. It is readily established by mathematical induction that (m + x) ≥ x m (x) for m ∈ N and x ≥ 0. Then and thus For α = ±1, b = a = 1, Theorem 5 leads to a dominant for the modified Bessel function obtained by Baricz in [21].
The final result uses the Chebyshev integral inequality [22, p. 40]: Suppose f and g are two integrable functions monotonic in the same sense (either both decreasing or both increasing). Let q : (a, b) → R be a positive integrable function.
The inequality in (23) is reversed if f and g are monotonic of the opposite sense.