ISSN 2291-8639
Volume 14, Number 2 (2017), 203-208
http://www.etamaths.com
INEQUALITIES FOR THE MODIFIED k-BESSEL FUNCTION
SAIFUL RAHMAN MONDAL1 AND KOTTAKKARAN SOOPPY NISAR2,∗
Abstract. The article considers the generalizedk-Bessel functions and represents it as Wright
func-tions. Then we study the monotonicity properties of the ratio of two different orders k- Bessel functions, and the ratio of thek-Bessel and thek-Bessel functions. The log-convexity with respect to the order of the k-Bessel also given. An investigation regarding the monotonicity of the ratio of thek-Bessel andk-confluent hypergeometric functions are discussed.
1. Introduction
One of the generalization of the classical gamma function Γ studied in [4] is defined by the limit formula
Γk(x) := lim n→∞
n!kn(nk)xk−1 (x)n,k
, k >0, (1.1)
where (x)n,k:=x(x+k)(x+2k). . .(x+(n−1)k) is calledk-Pochhammer symbol. The abovek−gamma
function also have an integral representation as
Γk(x) =
Z ∞
0
tx−1e−tkkdt, <(x)>0. (1.2)
Properties of the k-gamma functions have been studies by many researchers [6,8–11]. Following properties are required in sequel:
(i) Γk(x+k) =xΓk(x)
(ii) Γk(x) =k x k−1Γ x
k
(iii) Γk(k) = 1
(iv) Γk(x+nk) = Γk(x)(x)n,k
Motivated with the above generalization of the k-gamma functions, Romero et. al. [1] introduced thek−Bessel function of the first kind defined by the series
Jk,νγ,λ(x) := ∞ X
n=0
(γ)n, k
Γk(λn+υ+ 1)
(−1)n(x/2)n
(n!)2 , (1.3)
wherek∈R+;α, λ, γ, υ∈C; <(λ)>0 and<(υ)>0. They also established two recurrence relations
forJk,νγ,λ.
In this article, we are considering the following function:
Ik,νγ,λ(x) := ∞ X
n=0
(γ)n, k
Γk(λn+υ+ 1)
(x/2)n
(n!)2 , (1.4)
Since
lim
k,λ,γ→1I
γ,λ k,ν (x) =
∞ X
n=0 1 Γ (n+υ+ 1)
(x/2)n n! =
2
x ν2
Iν(
√
2x),
Received 18th March, 2017; accepted 25thMay, 2017; published 3rdJuly, 2017. 2010Mathematics Subject Classification. 33C10, 26D07.
Key words and phrases. generalizedk-Bessel functions; monotonicity; log-convexity; Tur´an type inequality.
c
2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
the classical modified Bessel functions of first kind. In this sense, we can call Ik,νγ,λ as the modified k-Bessel functions of first kind. In fact, we can express bothJk,νγ,λandIk,νγ,λtogether in
Wγ,λk,ν,c(x) := ∞ X
n=0
(γ)n, k
Γk(λn+ν+ 1)
(−c)n(x/2)n
(n!)2 , c∈R. (1.5)
We can termedWγ,λk,ν as the generalizedk-Bessel function.
First we study the representation formulas for Wγ,λk,ν in term of the classical Wright functions. Then
we will study about the monotonicity and log-convexity properties ofIk,νγ,λ.
2. Representation formula for the generalizedk-Bessel function
The generalized hypergeometric functionpFq(a1, . . . , ap;c1, . . . , cq;x), is given by the power series
pFq(a1, . . . , ap;c1, . . . , cq;z) =
∞ X
k=0
(a1)k· · ·(ap)k
(c1)k· · ·(cq)k(1)k
zk, |z|<1, (2.1)
where theci can not be zero or a negative integer. Herepor q or both are allowed to be zero. The
series (2.1) is absolutely convergent for all finitezifp≤qand for|z|<1 ifp=q+ 1. Whenp > q+ 1, then the series diverge forz6= 0 and the series does not terminate.
The generalized Wright hypergeometric functionpψq(z) is given by the series
pψq(z) =pψq
(ai, αi)1,p
(bj, βj)1,q
z = ∞ X k=0 Qp
i=1Γ(ai+αik)
j=1Γ(bj+βjk)
zk
k!, (2.2)
whereai, bj ∈C, and realαi, βj ∈R(i= 1,2, . . . , p;j= 1,2, . . . , q). The asymptotic behavior of this
function for large values of argument ofz∈Cwere studied in [13,14] and under the condition
q
X
j=1 βj−
p
X
i=1
αi>−1 (2.3)
in literature [18,19]. The more properties of the Wright function are investigated in [14–16].
Now we will give the representation of the generalizedk-Bessel functions in terms of the Wright and generalized hypergeometric functions.
Proposition 2.1. Let,k∈Randλ, γ, ν∈Csuch that <(λ)>0,<(ν)>0.Then
Wγ,λk,ν,c(x) = 1 kν+kk+1Γ γ
k
1ψ2
γ
k,1
ν+1 k , γ k (1,1) − cx
2kλk−1
Proof. Using the relations Γk(x) =k x k−1Γ x
k
and Γk(x+nk) = Γk(x)(x)n,k, the generalizedk-Bessel
functions defined in (1.5) can be rewrite as
Wγ,λk,ν,c(x) = ∞ X
n=0
Γk(γ+nk)
Γk(λn+ν+ 1)Γk(γ)
(−c)n
(n!)2 x 2 n (2.4) = 1
kν+kk+1Γ γ k
∞ X
n=0
Γ γk +n Γ λkn+ν+1k
Γ γk
(−c)n
Γ(n+ 1)Γ(n+ 1)
x
2kλk−1
n
(2.5)
= 1
kν+kk+1Γ γ k
1ψ2
γ
k,1
ν+1 k , γ k (1,1) − cx
2kλk−1
(2.6)
Hence the result follows.
3. Monotonicity and log-convexity properties
This section discuss the monotonicity and log-convexity properties for the modified k-Bessel func-tionsWγ,λk,ν,−1(x) =Iγ,λk,ν(x).
Lemma 3.1. [7] Consider the power series f(x) =P∞k=0akxk andg(x) =P∞k=0bkxk, whereak ∈R andbk>0 for allk. Further suppose that both series converge on|x|< r. If the sequence{ak/bk}k≥0
is increasing (or decreasing), then the function x7→ f(x)/g(x) is also increasing (or decreasing) on
(0, r).
The above lemma still holds when bothf andg are even, or both are odd functions.
Theorem 3.1. The following results holds true for the modifiedk-Bessel functions.
(1) For µ ≥ ν > −1, the function x 7→ Iγ,λk,µ(x)/Iγ,λk,ν(x) is increasing on (0,∞) for some fixed
k >0.
(2) If k ≥λ≥m > 0, the function x7→ Iγ,λk,ν(x)/Iγ,λm,ν(x) is increasing on (0,∞) for some fixed
ν >−1 andγ≥ν+ 1.
(3) The functionν 7→ Ik,νγ,λ(x) is log-convex on (0,∞) for some fixedk, γ >0 andx >0. Here, Ik,νγ,λ(x) := Γk(ν+ 1)Iγ,λk,ν(x).
(4) Suppose thatλ≥k >0 andν >−1. Then
(a) The function x7→Iγ,λk,ν(x)/Φk(a, c;x)is decreasing on(0,∞)fora≥c >0 and0< γ≤
ν+ 1. Here, Φk(a;c;x)is thek-confluent hypergeometric functions.
(b) The function x7→ Iγ,λk,ν(x)/Φk(γ;λ;x/2) is decreasing on (0,1) for γ > 0 and 0 < k ≤
λ≤ν+ 1.
(c) The function x7→Iγ,λk,ν(x)/Φk(γ;λ;x/2) is decreasing on[1,∞)for γ > 0 and 0< k ≤
min{λ, ν+ 1}.
Proof. (1) Form (1.4) it follows that
Iγ,λk,ν(x) = ∞ X
n=0
an(ν)xn and Iγ,λk,ν(x) =
∞ X
n=0
an(µ)xn,
where
an(ν) =
(γ)n,k
Γk(λn+ν+ 1)(n!)22n
and an(µ) =
(γ)n,k
Γk(λn+µ+ 1)(n!)22n
Consider the function
f(t) :=Γk(λt+µ+ 1) Γk(λt+ν+ 1)
.
Then the logarithmic differentiation yields
f0(t)
f(t) =λ(Ψk(λt+µ+ 1)−Ψk(λt+ν+ 1)). Here, Ψk= Γ0k/Γk is thek-digamma functions studied in [5] and defined by
Ψk(t) =
log(k)−γ1
k −
1 t +
∞ X
n=1 t
nk(nk+t) (3.1)
whereγ1is the Euler-Mascheronis constant. A calculation yields
Ψ0k(t) = ∞ X
n=0 1
(nk+t)2, k >0 and t >0. (3.2)
Clearly, Ψk is increasing on (0,∞) and hencef0(t)>0 for allt≥0 ifµ≥ν >−1. This, in particular,
implies that the sequence{dn}n≥0={an(ν)/an(µ)}n≥0 is increasing and hence the conclusion follows from Lemma3.1.
(2). This result also follows from Lemma 3.1 if the sequence {dn}n≥0 = {akn(ν)/amn(µ)}n≥0 is increasing fork≥m >0. Here,
akn(ν) = (γ)n,k Γk(λn+ν+ 1) (n!)
2 and a
m n (ν) =
(γ)n,m
Γm(λn+ν+ 1) (n!)
which together with the identity Γk(x+nk) = Γk(x)(x)n,k gives
dn =
(γ)n,k
(γ)n,m
Γm(λn+ν+ 1)
Γk(λn+ν+ 1)
= Γk(γ+nk) Γm(λn+ν+ 1) Γk(γ+nm) Γk(λn+ν+ 1)
.
Now to show that{dn}is increase, consider the function
f(y) := Γk(γ+yk) Γm(λy+ν+ 1) Γk(γ+ym) Γk(λy+ν+ 1)
The logarithmic differentiation off yields f0(y)
f(y) =kΨk(γ+yk) +λΨm(λy+ν+ 1)−mΨm(γ+ym)−λΨk(λy+ν+ 1) (3.3) Ifγ≥ν+ 1 andk≥λ≥m, then (3.3) can be rewrite as
f0(y)
f(y) ≥λ Ψk(ν+ 1 +yk)−Ψk(λy+ν+ 1)
+m Ψm(λy+ν+ 1)−Ψm(ν+ 1 +ym)
≥0. (3.4)
This conclude thatf, and consequently the sequence{dn}n≥0, is increasing. Finally the result follows
from the Lemma3.1.
(3). It is known that sum of the log-convex functions is log-convex. Thus, to prove the result it is enough to show that
ν7→akn(ν) := (γ)n,kΓk(ν+ 1) Γk(λn+ν+ 1) (n!)
2
is log-convex.
A logarithmic differentiation of an(ν) with respect toν yields
∂ ∂νlog a
k n(ν)
= Ψk(ν+ 1)−Ψk(λn+ν+ 1).
This along with (3.2) gives
∂2 ∂ν2log a
k n(ν)
= Ψ0k(ν+ 1)−Ψ0k(λn+ν+ 1)
= ∞ X
r=0
1
(rk+ν+ 1)2 − ∞ X
r=0
1
(rk+λn+ν+ 1)2
= ∞ X
r=0
λn(2rk+λn+ 2ν+ 2)
(rk+ν+ 1)2(rk+λn+ν+ 1)2 >0,
for alln≥0,k >0 andν >−1. Thus, ν7→ak
n(ν) is log-convex and hence the conclusion.
(4). Denote Φk(a, c;x) =P∞n=0cn,k(a, c)xn andIγ,λk,ν(x) =P∞n=0an(ν)xn,where
an(ν) =
(γ)n,k
Γk(λn+ν+ 1)(n!)22n
and dn,k(a, c) =
(a)n,k
(c)n,kn!
withv >−1 anda, c, λ, γ, k >0.To apply Lemma 3.1, consider the sequence{wn}n≥0defined by
wn=
an(ν)
dn,k(a, c)
= Γk(γ+nk) 2nΓ
k(γ) Γk(λn+α+ 1) (n!)
2.
Γk(a) Γk(c+nk)n!
Γk(a+nk) Γk(c)
= Γk(a) Γk(γ) Γk(c)
ρk(n)
where
ρk(x) =
Γk(γ+xk) Γk(c+xk)
Γk(λx+ν+ 1) Γk(a+xk) 2xΓ(x+ 1)
In view of the increasing properties of Ψk on (0,∞), and
ρ0(x)
ρ(x) =kψk(γ+xk) +kψk(c+xk)−λψk(λx+α+ 1)−kψk(a+xk),
it follows that fora≥c >0,λ≥kandν+ 1≥γ, the functionρis decreasing on (0,∞) and thus the sequence{wn}n≥0 also decreasing. Finally the conclusion for (a) follows from the Lemma 3.1.
In the case (b) and (c), the sequence{wn}reduces to
wn=
an(ν)
dn,k(γ, λ)
= ρk(n) Γk(λ)
where
ρk(x) =
Γk(λ+xk)
Γk(ν+ 1 +λx)Γ(x+ 1)
.
Now as in the proof of part (a)
ρ0k(x) ρk(x)
=kΨk(λ+xk)−λΨk(ν+ 1 +xk)−Ψ(x+ 1)>0,
ifν+ 1 +λx≥λ+xk. Now forx∈(0,1), this inequality holds if 0< k≤λ≤ν+ 1, while forx≥1,
it is required thatk≤min{λ, ν+ 1}.
References
[1] L.G. Romero, G.A.Dorrego and R.A. Cerutti, The k-Bessel function of first kind, Int. Math. Forum, 38(7)(2012), 1859–1854.
[2] GN. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library Edition, Cambridge University Press, Cambridge (1995). Reprinted (1996)
[3] A. Erd´elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, I, II, McGraw-Hill Book Company, Inc., New York, 1953. New York, Toronto, London, 1953.
[4] R. Diaz and E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat. 15(2) (2007), 179–192.
[5] K. Nantomah, E. Prempeh, Some Inequalities for the k-Digamma Function, Math. Aeterna, 4(5) (2014), 521–525. [6] S. Mubeen, M. Naz and G. Rahman, A note on k-hypergemetric differential equations, J. Inequal. Spec. Funct.
4(3) (2013), 8–43.
[7] M. Biernacki and J. Krzy ˙z, On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sk lodowska. Sect. A. 9 (1957), 135–147.
[8] C. G. Kokologiannaki, Properties and inequalities of generalizedk-gamma, beta and zeta functions, Int. J. Con-temp. Math. Sci. 5(13-16) (2010), 653–660.
[9] C. G. Kokologiannaki and V. Krasniqi, Some properties of thek-gamma function, Matematiche (Catania), 68(1) (2013), 13–22.
[10] V. Krasniqi, A limit for thek-gamma andk-beta function, Int. Math. Forum, 5(33-36) (2010), 1613–1617. [11] M. Mansour, Determining thek-generalized gamma function Γk(x) by functional equations, Int. J. Contemp. Math.
Sci., 4(21-24) (2009), 1037–1042.
[12] G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge Univ. Press, Cambridge, 1999.
[13] C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc., 27(4) (1928), 389–400.
[14] A. A. Kilbas, M. Saigo and J. J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal. 5(4) (2002), 437–460.
[15] A. A. Kilbas and N. Sebastian, Generalized fractional integration of Bessel function of the first kind, Integral Transforms Spec. Funct. 19 (11-12) (2008), 869–883.
[16] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
[17] E. D. Rainville, Special functions, Macmillan, New York, 1960.
[18] E. M. Wright, The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London, Ser. A. 238 (1940), 423–451.
[19] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. (2) 46(1940), 389–408.
1
Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf, Kingdom of Saudi Arabia
2
∗
