International Journal of Analysis and Applications
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 generalized k-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 the k-Bessel and the k-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
the k-Bessel and k-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)
x
k
−1

(x)n,k
, k > 0, (1.1)

where (x)n,k := x(x+k)(x+2k) . . . (x+(n−1)k) is called k-Pochhammer symbol. The above k−gamma
function also have an integral representation as

Γk(x) =

∫ ∞
0

tx−1e−
tk

k dt, <(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
the k−Bessel function of the first kind defined by the series

J
γ,λ
k,ν (x) :=

∞∑
n=0

(γ)n, k
Γk (λn + υ + 1)

(−1)n (x/2)n

(n!)
2

, (1.3)

where k ∈ R+; α,λ,γ,υ ∈ C; <(λ) > 0 and <(υ) > 0. They also established two recurrence relations
for J

γ,λ
k,ν .

In this article, we are considering the following function:

I
γ,λ
k,ν (x) :=

∞∑
n=0

(γ)n, k
Γk (λn + υ + 1)

(x/2)
n

(n!)
2
, (1.4)

Since

lim
k,λ,γ→1

I
γ,λ
k,ν (x) =

∞∑
n=0

1

Γ (n + υ + 1)

(x/2)
n

n!
=

(
2

x

)ν
2

Iν(
√

2x),

Received 18th March, 2017; accepted 25th May, 2017; published 3rd July, 2017.
2010 Mathematics Subject Classification. 33C10, 26D07.
Key words and phrases. generalized k-Bessel functions; monotonicity; log-convexity; Turán 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.

203



204 MONDAL AND NISAR

the classical modified Bessel functions of first kind. In this sense, we can call I
γ,λ
k,ν as the modified

k-Bessel functions of first kind. In fact, we can express both J
γ,λ
k,ν and I

γ,λ
k,ν together in

W
γ,λ
k,ν,c(x) :=

∞∑
n=0

(γ)n, k
Γk(λn + ν + 1)

(−c)n(x/2)n

(n!)
2

, c ∈ R. (1.5)

We can termed W
γ,λ
k,ν as the generalized k-Bessel function.

First we study the representation formulas for W
γ,λ
k,ν in term of the classical Wright functions. Then

we will study about the monotonicity and log-convexity properties of I
γ,λ
k,ν .

2. Representation formula for the generalized k-Bessel function

The generalized hypergeometric function pFq(a1, . . . ,ap; c1, . . . ,cq; x), is given by the power series

pFq(a1, . . . ,ap; c1, . . . ,cq; z) =

∞∑
k=0

(a1)k · · ·(ap)k
(c1)k · · ·(cq)k(1)k

zk, |z| < 1, (2.1)

where the ci can not be zero or a negative integer. Here p or q or both are allowed to be zero. The
series (2.1) is absolutely convergent for all finite z if p ≤ q and for |z| < 1 if p = q + 1. When p > q + 1,
then the series diverge for z 6= 0 and the series does not terminate.

The generalized Wright hypergeometric function pψq(z) is given by the series

pψq(z) = pψq

[
(ai,αi)1,p
(bj,βj)1,q

∣∣∣∣z
]

=

∞∑
k=0

∏p
i=1 Γ(ai + αik)∏q
j=1 Γ(bj + βjk)

zk

k!
, (2.2)

where ai,bj ∈ C, and real αi,βj ∈ R (i = 1, 2, . . . ,p; j = 1, 2, . . . ,q). The asymptotic behavior of this
function for large values of argument of z ∈ C were studied in [13, 14] and under the condition

q∑
j=1

βj −
p∑
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 generalized k-Bessel functions in terms of the Wright and

generalized hypergeometric functions.

Proposition 2.1. Let, k ∈ R and λ,γ,ν ∈ C such that <(λ) > 0,<(ν) > 0. Then

W
γ,λ
k,ν,c(x) =

1

k
ν+k+1
k Γ

(
γ
k

)1ψ2 [ (γk , 1)(ν+1
k
, γ
k

)
(1, 1)

∣∣∣∣− cx
2k

λ
k
−1

]
Proof. Using the relations Γk (x) = k

x
k
−1Γ

(
x
k

)
and Γk (x + nk) = Γk(x)(x)n,k, the generalized k-Bessel

functions defined in (1.5) can be rewrite as

W
γ,λ
k,ν,c(x) =

∞∑
n=0

Γk(γ + nk)

Γk(λn + ν + 1)Γk(γ)

(−c)n

(n!)2

(x
2

)n
(2.4)

=
1

k
ν+k+1
k Γ

(
γ
k

) ∞∑
n=0

Γ
(
γ
k

+ n
)

Γ
(
λ
k
n + ν+1

k

)
Γ
(
γ
k

) (−c)n
Γ(n + 1)Γ(n + 1)

(
x

2k
λ
k
−1

)n
(2.5)

=
1

k
ν+k+1
k Γ

(
γ
k

)1ψ2 [ (γk , 1)(ν+1
k
, γ
k

)
(1, 1)

∣∣∣∣− cx
2k

λ
k
−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-

tions W
γ,λ
k,ν,−1(x) = I

γ,λ
k,ν (x).

Following lemma due to Biernacki and Krzyż [7] will be required.



INEQUALITIES FOR THE MODIFIED k-BESSEL FUNCTION 205

Lemma 3.1. [7] Consider the power series f(x) =
∑∞
k=0 akx

k and g(x) =
∑∞
k=0 bkx

k, where ak ∈ R
and bk > 0 for all k. Further suppose that both series converge on |x| < r. If the sequence {ak/bk}k≥0
is increasing (or decreasing), then the function x 7→ f(x)/g(x) is also increasing (or decreasing) on
(0,r).

The above lemma still holds when both f and g are even, or both are odd functions.

Theorem 3.1. The following results holds true for the modified k-Bessel functions.

(1) For µ ≥ ν > −1, the function x 7→ Iγ,λk,µ(x)/I
γ,λ
k,ν (x) is increasing on (0,∞) for some fixed

k > 0.
(2) If k ≥ λ ≥ m > 0, the function x 7→ Iγ,λk,ν (x)/I

γ,λ
m,ν(x) is increasing on (0,∞) for some fixed

ν > −1 and γ ≥ ν + 1.
(3) The function ν 7→ Iγ,λk,ν (x) is log-convex on (0,∞) for some fixed k,γ > 0 and x > 0. Here,
Iγ,λk,ν (x) := Γk(ν + 1)I

γ,λ
k,ν (x).

(4) Suppose that λ ≥ k > 0 and ν > −1. Then
(a) The function x 7→ Iγ,λk,ν (x)/Φk (a,c; x) is decreasing on (0,∞) for a ≥ c > 0 and 0 < γ ≤

ν + 1. Here, Φk (a; c; x) is the k-confluent hypergeometric functions.

(b) The function x 7→ 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 x 7→ 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

I
γ,λ
k,ν (x) =

∞∑
n=0

an(ν)x
n and I

γ,λ
k,ν (x) =

∞∑
n=0

an(µ)x
n,

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

f′(t)

f(t)
= λ(Ψk(λt + µ + 1) − Ψk(λt + ν + 1)).

Here, Ψk = Γ
′
k/Γk is the k-digamma functions studied in [5] and defined by

Ψk(t) =
log(k) −γ1

k
−

1

t
+

∞∑
n=1

t

nk(nk + t)
(3.1)

where γ1 is the Euler-Mascheronis constant.
A calculation yields

Ψ′k(t) =

∞∑
n=0

1

(nk + t)2
, k > 0 and t > 0. (3.2)

Clearly, Ψk is increasing on (0,∞) and hence f′(t) > 0 for all t ≥ 0 if µ ≥ 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 Lemma 3.1.

(2). This result also follows from Lemma 3.1 if the sequence {dn}n≥0 = {akn(ν)/amn (µ)}n≥0 is
increasing for k ≥ m > 0. Here,

akn (ν) =
(γ)n,k

Γk (λn + ν + 1) (n!)
2

and amn (ν) =
(γ)n,m

Γm (λn + ν + 1) (n!)
2
,



206 MONDAL AND NISAR

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 of f yields

f′(y)

f(y)
= kΨk(γ + yk) + λΨm (λy + ν + 1) −mΨm(γ + ym) −λΨk (λy + ν + 1) (3.3)

If γ ≥ ν + 1 and k ≥ λ ≥ m, then (3.3) can be rewrite as
f′(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 that f, and consequently the sequence {dn}n≥0, is increasing. Finally the result follows
from the Lemma 3.1.

(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

ν 7→ akn (ν) :=
(γ)n,k Γk (ν + 1)

Γk (λn + ν + 1) (n!)
2

is log-convex.
A logarithmic differentiation of an(ν) with respect to ν yields

∂

∂ν
log
(
akn (ν)

)
= Ψk (ν + 1) − Ψk (λn + ν + 1) .

This along with (3.2) gives

∂2

∂ν2
log
(
akn (ν)

)
= Ψ′k (ν + 1) − Ψ

′
k (λn + ν + 1)

=

∞∑
r=0

1

(rk + ν + 1)2
−
∞∑
r=0

1

(rk + λn + ν + 1)2

=
∞∑
r=0

λn(2rk + λn + 2ν + 2)

(rk + ν + 1)2(rk + λn + ν + 1)2
> 0,

for all n ≥ 0, k > 0 and ν > −1. Thus, ν 7→ akn (ν) is log-convex and hence the conclusion.

(4). Denote Φk (a,c; x) =
∑∞
n=0 cn,k(a,c)x

n and I
γ,λ
k,ν (x) =

∑∞
n=0 an(ν)x

n, where

an(ν) =
(γ)n,k

Γk(λn + ν + 1)(n!)22n
and dn,k (a,c) =

(a)n,k
(c)n,k n!

with v > −1 and a,c,λ,γ,k > 0. To apply Lemma 3.1, consider the sequence {wn}n≥0 defined by

wn =
an (ν)

dn,k (a,c)
=

Γk (γ + nk)

2nΓk (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)
.



INEQUALITIES FOR THE MODIFIED k-BESSEL FUNCTION 207

In view of the increasing properties of Ψk on (0,∞), and
ρ′ (x)

ρ (x)
= kψk (γ + xk) + kψk (c + xk) −λψk (λx + α + 1) −kψk (a + xk) ,

it follows that for a ≥ c > 0, λ ≥ k and ν + 1 ≥ γ, the function ρ is 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)

ρ′k (x)

ρk (x)
= kΨk(λ + xk) −λΨk(ν + 1 + xk) − Ψ(x + 1) > 0,

if ν + 1 + λx ≥ λ + xk. Now for x ∈ (0, 1), this inequality holds if 0 < k ≤ λ ≤ ν + 1, while for x ≥ 1,
it is required that k ≤ min{λ,ν + 1}. �

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1 Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf, Kingdom

of Saudi Arabia

2 Department of Mathematics, College of Arts & Science-Wadi Al-dawaser, Prince Sattam bin Abdulaziz
University, Alkharj, Kingdom of Saudi Arabia



208 MONDAL AND NISAR

∗Corresponding author: n.sooppy@psau.edu.sa, ksnisar1@gmail.com


	1. Introduction
	2. Representation formula for the generalized k-Bessel function
	3. Monotonicity and log-convexity properties
	References