On the geometric properties of the Bessel-Struve kernel function

0.14 MB·

by  Saiful R Mondal

ON THE GEOMETRIC PROPERTIES OF THE
BESSEL-STRUVE KERNEL FUNCTION
6
1 ∗
SAIFUL R. MONDAL
0
2
n Abstract. This paper introduce the Bessel-Struve kernel func-
a tions ν definedonthe unitdiskinthecomplexplane. We studies
J B
9 ttihoensc.loSsue-ffitoc-iceonntvceoxnitdyitoiofnBoνnwνitfhorrweshpiecchttthoesfeuvnecrtailonstzarlνikeisfsutnacr--
2 B
like is given.

]
V
C
. 1. Introduction and Preliminaries
h
t
a 1.1. Bessel-Struve Kernel functions. The function Jν, known as
m
the Bessel function of the first kind of order ν, is a particular solution
[ of the second order Bessel differential equation

1 z2y′′(z)+zy′(z)+(z2 ν2)y(z) = 0.
v
−
2
This function has the power series representation
0
1 ∞ ( 1)n z 2n+ν
8
J (z) = − , z < . (1.1)
0 ν n!Γ(ν +n+1) 2 | | ∞
. Xn=0 (cid:16) (cid:17)
1
0 On the other hand the modified Bessel function I (z) is the particular
ν
6 solution of the differential equation
1
: z2y′′(z)+zy′(z) (z2 ν2)y(z) = 0,
v
− −
i
X and have the series representation
r ∞
a 1 z 2n+ν
I (z) = , z < . (1.2)
ν
n!Γ(ν +n+1) 2 | | ∞
Xn=0 (cid:16) (cid:17)
The Struve function of order ν given by
∞ ( 1)k z 2n+ν+1
H (z) := − (1.3)
ν Γ n+ν + 3 Γ n+ 3 2
Xn=0 2 2 (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
2010 Mathematics Subject Classification. 30C45, 33C10, 30C80,40G05.
Key words and phrases. Besselfunctions,Struvefunctions,Bessel-Struvekernel,
Starlike, Close-to-convex.
TheauthorthankstheDeanshipofScientificResearchatKingFaisalUniversity
∗
for funding his work under project number 150244.
1
2 SAIFULR.MONDAL
isaparticularsolutionofthenon-homogeneousBessel differentialequa-
tion
4 z ν+1
z2y′′(z)+zy′(z)+(z2 ν2)y(z) = 2 . (1.4)
− √π (cid:0)Γ (cid:1)ν + 1
2
(cid:0) (cid:1)
A solution of the non-homogeneous modified Bessel equation
4 z ν+1
z2y′′(z)+zy′(z) (z2 +ν2)y(z) = 2 . (1.5)
− √π (cid:0)Γ (cid:1)ν + 1
2
(cid:0) (cid:1)
yields the modified Struve function
L (z) := ie−iν2π H (iz)
ν ν
−
∞
1 z 2n+ν+1
= . (1.6)
Γ n+ν + 3 Γ n+ 3 2
Xn=0 2 2 (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
The Struve functions occur in areas of physics and applied mathemat-
ics, for example, in water-wave and surface-wave problems [3], as well
as in problems on unsteady aerodynamics [15]. The Struve functions
are also important in particle quantum dynamical studies of spin de-
coherence [14] and nanotubes [13].
Consider the Bessel-Struve kernel function defined on the unit
ν
disk D = z : z < 1 as B
{ | | }
1
(z) := j (iz) ih (iz), ν > , (1.7)
ν ν ν
B − −2
where, j (z) := 2νz−νΓ(ν +1)J (z) and h (z) := 2νz−νΓ(ν +1)H (z)
ν ν ν ν
are respectively known as the normalized Bessel functions and the
normalized Struve functions of first kind of index ν. The Bessel-
Struve transformation and Bessel-Struve kernel functions are appeared
in many article. In [10], Hamem et. al. studies an analogue of the
Cowling’s Price theorem for the BesselStruve transform defined on
real domain and also provide Hardy type theorem associated with this
transform. The Bessel-Struve intertwining operator on C is considered
in [8, 11]. The fock space of the Bessel-Struve kernel functions is dis-
cussed in [9]. The kernel z (γz), γ C is the unique solution of
ν
7→ B ∈
the initial value problem
λΓ(ν +1)
u(z) = λ2u(z), u(0) = 1,u′(0) = . (1.8)
Lν √πΓ(ν + 3)
2
Here , ν > 1/2 is the Bessel-Struve operator given by
ν
L −
d2u 2ν +1 du du
(u(z)) := (z)+ (z) (0) . (1.9)
ν
L dz2 z (cid:18)dz − dz (cid:19)
ON THE GEOMETRIC PROPERTIES OF THE BESSEL-STRUVE KERNEL FUNCTION3
Now from (1.1) and (1.6), it is evident that (taking γ = 1) possesses
ν
B
the power series
∞ Γ(ν +1)Γ n+1
(z) := 2 zn. (1.10)
Bν √πn!Γ n +(cid:0)ν +(cid:1)1
Xn=0 2
(cid:0) (cid:1)
The kernel also have the integral representation
ν
B
(z) := 2Γ(ν +1) 1(1 t2)ν−12eztdt. (1.11)
Bν √πΓ ν + 1 Z −
2 0
(cid:0) (cid:1)
The identity (1.8) and (1.9) together imply that satisfy the differ-
ν
B
ential equation
z2g′′(z)+(2ν +1)zg′(z) zg (z) = zM, (1.12)
ν ν − ν
where M = 2Γ(ν +1) √π Γ(ν + 1) −1.
2
Another significance(cid:0) is that ν c(cid:1)an be express as the sum of the
B
modified Bessel and the modified Struve function of first kind of order
ν. For the sake of completeness, in the following result we established
this relation.
Proposition 1.1. For ν > 0, the following identity holds:
zν (z) = 2νΓ(ν +1)(I (z)+L (z)).
ν ν ν
B
Thefunction havethefollowingrecurrencerelationwhichisuseful
ν
B
in sequel.
Proposition 1.2. For ν > 0, the following recurrence relation holds
for :
ν
B
z ′(z) = 2ν (z) 2ν (z). (1.13)
Bν Bν−1 − Bν
1.2. Starlike and close-to-convex functions. LetD = z : z < 1
{ | | }
be the unit disk and be the class of all analytic functions f defined
on D such that f(0)A= 0 = f′(0) 1. Clearly each f have the
− ∈ A
power series
∞
f(z) = z + a zn. (1.14)
n
Xn=2
A function f is said to be starlike if f(D) is starlike with respect
∈ A
to the origin. Now if for any starlike function g and for some real
number β, we have Re(eiβf′(z)/g(z)) > 0, then the function f is said
tobeclose-to-convex with respect to thestarlike function g. Afunction
4 SAIFULR.MONDAL
f is convex if f(D) is convex. The starlike and convex functions
∈ A
can be represent analytically as
zf′(z) zf′′(z)
Re > 0 and Re 1+ > 0,
(cid:18) f(z) (cid:19) (cid:18) f′(z) (cid:19)
respectively. Traditionally the class of starlike functions is denoted as
∗, while the class of close-to-convex, and convex functions aredenoted
S
respectively as and . These classes also be generalized by order
C K
λ [0,1) with the analytical formulation as follows:
∈
zf′(z)
f ∗(λ) Re > λ,
∈ S ⇔ (cid:18) f(z) (cid:19)
zf′′(z)
f (λ) Re 1+ > λ,
∈ C ⇔ (cid:18) f′(z) (cid:19)
zf′(z)
f (λ) Re > λ, for some g ∗.
∈ K ⇔ (cid:18) g(z) (cid:19) ∈ S
AccordingtotheAlexanderdualitytheorem[4], thefunctionf : D C
→
isin (ν)ifandonlyifz zf′(z)isstarlikeoforderν. Hereweremark
C →
that the definition of (ν) is also valid for non-normalized analytic
function f : D C witCh the property f′(0) = 0.
→ 6
Let introduce another subclass of ∗(λ) consisting of functions f for
S
which
zf′(z)
1 < 1 λ, (1.15)
(cid:12) f(z) − (cid:12) −
(cid:12) (cid:12)
(cid:12) (cid:12)
and denoted the class as (λ). The Alexander duality theorem can
(cid:12) 1 (cid:12)
S
be apply to the class (λ) and a function f is said to be in the class
1
S
(λ) if zf′(z) (λ).
1 1
C ∈ S
Following result is required in sequel.
Lemma 1.1. [16] Let λ [0,1/2] be fixed and β 0. If f and
∈ ≥ ∈ A
zf′(z) 1 1−β zf′′(z) β < (1 λ)1−2β 1 3λ +λ2 β, (1.16)
f(z) − f′(z) − − 2
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:0) (cid:1)
for all(cid:12)z D, th(cid:12)en f(cid:12) ∗((cid:12)λ).
(cid:12) (cid:12) (cid:12) (cid:12)
∈ ∈ S
Next we state our main results which are proved in Section 2 by
using Lemma 1.1.
Theorem 1.1. Let the function as defined in (1.8) satisfy the in-
ν
B
equality
z ′(z)
Bν < 1 λ, (1.17)
(cid:12) (z) (cid:12) −
(cid:12) Bν (cid:12)
for λ [0,1/2]. Then z (cid:12) ∗(λ)(cid:12).
ν(cid:12) (cid:12)
∈ B ∈ S
ON THE GEOMETRIC PROPERTIES OF THE BESSEL-STRUVE KERNEL FUNCTION5
Now we will introduce a subclass of ∗(λ) consisting of functions f
S
satisfying the inequality
zf′(z)
1 < 1 λ,
(cid:12) f(z) − (cid:12) −
(cid:12) (cid:12)
(cid:12) (cid:12)
is known as the class of(cid:12)the starlik(cid:12)e functions with respect to 1 and
denoted as ∗(ν).
S1
Inour next result, we obtainsufficient condition by which theBessel-
Struve kernel functions is starlike with respect to 1.
Theorem 1.2. Let the function as defined in (1.8) satisfy the in-
ν
B
equality
z ′′(z) 2 3λ+2λ2
Bν < − , (1.18)
(cid:12) ′(z) (cid:12) 2(1 λ)
(cid:12) Bν (cid:12) −
(cid:12) (cid:12)
for λ ∈ [0,1/2]. Then (cid:12)Bν ∈ S1∗(cid:12)(λ).
Following problem is well known in the literature:
Problem 1.1. Find the conditions on a such that
n
∞
f(z) = z + a zn (1.19)
n
Xn=2
is close-to-convex, or starlike or convex or any other subclasses of uni-
valent functions.
Now in accordance with the Problem 1.1, we need to find the condi-
tion on ν such that the Bessel-Struve kernel functions or it’s normal-
ization, will be in any of the classes mention above.
There are many results in the literature (see. [2, 5, 12, 17] and
referencetheirin)whichanswertheaboveproblem. Asperrequirement
for this work, we listed few of them here. Here we would like to remark
that the functions
z z z z
z, , , and ,
1 z 1 z2 (1 z)2 1 z +z2
− − − −
and their particular rotations
z z z z
, , and
1+z 1+z2 (1+z)2 1+z +z2
are the only nine functions which have integer coefficients and are star-
like univalent in D (See [7]). The sufficient coefficient conditions for
which a function f is close-to-convex can be easily obtain atleast
∈ A
when the corresponding starlike functions is one of the above listed
form. In this article we will consider for z, z/(1 z) and z/(1 z2).
− −
6 SAIFULR.MONDAL
Lemma 1.2. [2] Let a ∞ be a sequence of non-negative real num-
{ n}n=1
bers such that a = 1, ∆a 0 when n 1 and ∆2a when n 2.
1 n n
≥ ≥ ≥
Then the function f, defined in (1.19), is starlike and close-to-convex
with respect to the starlike functions z and z/(1 z). Here ∆a =
n
−
na (n+1)a and ∆m+1a = ∆m(∆a ), m = 1,2, .
n n+1 n n
− ···
The starlikeness and close-to-convexity of z is obtained by using
ν
B
Lemma 1.2 in the following result.
Theorem 1.3. For ν 1/2, the normalized Bessel-Struve kernel func-
≥
tion z is starlike. The function z is also close-to-convex with re-
ν ν
B B
spect to the starlike functions z and z/(1 z).
−
It can be observed that Theorem 1.3 can also be proved by using the
following lemma given in [12].
Lemma 1.3. [12] Let a be a sequence of positive real number
n n≥1
{ }
such that a 2a 6a and n(n 2)a (n 1)(n + 1)a for
1 2 3 n n+1
n 3. Then≥f(z) =≥z + ∞ a zn−is close≥-to-co−nvex with respect to
≥ n=2 n
both the starlike functionsPz and z/(1 z). Further, the function f is
starlike univalent in D. −
In our next result we will study the close-to-convexity of z with
ν
B
respect to the starlike functions z/(1 z2).
−
Theorem 1.4. Ifν ν 19.6203, the function z is close-to-convex
0 ν
≥ ≈ B
with respect to the starlike functions z/(1 z2).
−
The following result is use to prove Theorem 1.4.
Lemma 1.4. [12, Theorem 4.4] Let a be a sequence of positive
n n≥1
{ }
real numbers such that a = 1. Suppose that a 8a , and (n 1)a
1 1 2 n
(n+1)a for all n 2. Then the function f(≥z) = z + ∞−a zn ≥is
n+1 ≥ n=2 n
close-to-convex with respect to the starlike functions z/(1Pz2).
−
2. Proof of the main results
Proof of Proposition 1.1. From (1.10), it follows that
∞ Γ(ν +1)Γ n+1
zν (z) = 2 zn+ν (2.20)
Bν √πn!Γ n +(cid:0)ν +(cid:1)1
Xn=0 2
∞ Γ(ν +(cid:0) 1)Γ m+(cid:1)1
= 2 z2m+ν
√π(2m)!Γ(m(cid:0) +ν +(cid:1)1)
mX=0
∞
Γ(ν +1)Γ(m+1)
+ z2m+1+ν.
√π(2m+1)!Γ m+ν + 3
mX=0 2
(cid:0) (cid:1)
ON THE GEOMETRIC PROPERTIES OF THE BESSEL-STRUVE KERNEL FUNCTION7
The Legendre duplication formula (see [1, 6])
1
Γ(z)Γ z + = 21−2z √π Γ(2z)
(cid:18) 2(cid:19)
shows that
Γ m+ 1 1 Γ(m+1) 1
2 = and = .
√(cid:0)π(2m)(cid:1)! 22mm! √π(2m+1)! 22m+1Γ m+ 3
2
(cid:0) (cid:1)
This along with (1.2) and (1.6), the identity (2.20) reduce to
∞ z 2m+ν z 2m+ν
zν (z) = 2νΓ(ν +1) 2
Bν m!Γ(cid:0)(m(cid:1)+ν +1) 2
mX=0 (cid:16) (cid:17)
∞ z 2m+ν+1
+ 2
Γ m+(cid:0)3 (cid:1)Γ m+ν + 3
mX=0 2 2
= 2νΓ(ν +(cid:0)1)(I (z(cid:1))+(cid:0)L (z)). (cid:1)
ν ν
(cid:3)
This complete the proof.
Proof of Proposition 1.2. Differentiatingtheseries(1.10),itfollowsthat
d ∞ nΓ(ν +1)Γ n+1
z (z) = 2 zn
dzBν √πn! Γ n +(cid:0)ν +1(cid:1)
Xn=0 2
∞ n +(cid:0)ν Γ(ν)Γ (cid:1)n+1
= 2ν 2 2 zn
(cid:0)√πn! Γ(cid:1) n +ν(cid:0)+1(cid:1)
Xn=0 2
∞ Γ(ν(cid:0)+1)Γ n+1(cid:1)
2ν 2 zn
− √πn! Γ n +(cid:0)ν +(cid:1)1
Xn=0 2
= 2ν (z) 2ν (cid:0)(z). (cid:1) (cid:3)
ν−1 ν
B − B
Proof of Theorem 1.1. Denote f(z) = z (z). Then a computation
ν
B
together with the hypothesis (1.18) yield
zf′(z) z ′(z)
1 = Bν < 1 λ,
(cid:12) f(z) − (cid:12) (cid:12) (z) (cid:12) −
(cid:12) (cid:12) (cid:12) Bν (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
which is equivalent(cid:12) to (1.16) (cid:12)for β(cid:12) = 0. T(cid:12) he conclusion follows from
(cid:3)
Lemma 1.1.
Proof of Theorem 1.2. Define h : D as
→ C
2Γ(ν +3/2)
h(z) := ( (z) 1). (2.21)
ν
Γ(ν +1) B −
8 SAIFULR.MONDAL
Then clearly h . Now a computation yield
∈ A
zh′′(z) z ′′(z) 2 3ν +2ν2
= Bν < − .
(cid:12) h′(z) (cid:12) (cid:12) ′(z) (cid:12) 2(1 ν)
(cid:12) (cid:12) (cid:12) Bν (cid:12) −
(cid:12) (cid:12) (cid:12) (cid:12)
Taking β = 1, fro(cid:12)m Lem(cid:12)ma (cid:12)1.1, it fo(cid:12)llows that h ∗(λ) with respect
∈ S
to origin. Now Theorem 1.2 follows from the definition of h in (2.21).
(cid:3)
Proof of Theorem 1.3. From (1.10), we can express z as
ν
B
∞
z (z) = a zn, (2.22)
ν n
B
Xn=1
where
Γ(ν +1)Γ n
a = 1 and a = 2 , n 2. (2.23)
1 n √π(n 1)! Γ (cid:0)n+1(cid:1)+ν ≥
− 2
(cid:0) (cid:1)
Define the function g : [0, ) R as
n
∞ →
Γ n +1+ν
g (ν) := 2 . (2.24)
n Γ(cid:0) n+1 +ν (cid:1)
2
(cid:0) (cid:1)
The logarithmic differentiation with respect to ν implies
n n+1
g′(ν) = g (ν) Ψ +ν +1 Ψ +ν . (2.25)
n n (cid:18) 2 − (cid:18) 2 (cid:19)(cid:19)
(cid:16) (cid:17)
Here Ψ is the well-known digamma functions which is an increasing
function on [0, ), and consequently g is also increasing. Thus for
n
∞
ν 1/2,
≥
Γ n+3
g (ν) g (1/2) = 2 (2.26)
n ≥ n Γ (cid:0)n +1(cid:1)
2
(cid:0) (cid:1)
Thus for n 1 and ν 1/2, it follows that
≥ ≥
a Γ(ν +1)Γ n √π(n)!Γ n+2 +ν
n = 2 2 (2.27)
a √π(n 1)!Γ n(cid:0)+1(cid:1)+ν × Γ(ν +1(cid:0))Γ n+1 (cid:1)
n+1 − 2 2
nΓ n Γ n+(cid:0)3 (cid:1) (cid:0) (cid:1)
2 2 = n+1.
≥ Γ n+(cid:0)1 (cid:1)Γ (cid:0)n +(cid:1)1
2 2
(cid:0) (cid:1) (cid:0) (cid:1)
This implies for n 1
≥
∆a = na (n+1)a (n2 1)a 0,
n n n+1 n+1
− ≥ − ≥
ON THE GEOMETRIC PROPERTIES OF THE BESSEL-STRUVE KERNEL FUNCTION9
and for n 2 we have
≥
∆2a = na 2(n+1)a +(n+2)a
n n n+1 n+2
−
(n+1)(n 2)a +(n+2)a
n+1 n+2
≥ −
(n+2)(n2 n 1)a > 0.
n+2
≥ − −
Thus a satisfy the hypothesis of Lemma 1.2 and hence the conclu-
n
{ } (cid:3)
sion.
Proof of Theorem 1.4. The inequality (2.27) yield that for n 2, and
≥
ν 1/2.
≥
(n 1)a (n+1)a na > 0,
n n+1 n+1
− − ≥
Now from (2.23) it follows that the coefficient a satisfy the hypothesis
n
a 8a is equivalent to √πΓ(ν +3/2) 8Γ(ν +1) which holds when
1 2
≥ ≥
ν ν , where ν 19.6203 is the positive root of the identity
0 0
≥ ≈
√πΓ(ν +3/2) = 8Γ(ν +1).
Now the result follows from the Lemma 1.4. (cid:3)
Problem 2.1 (Open). Find the sharp lowest bound for ν > 1 so that
z is starlike in D and also close-to-convex with respect to t−he starlike
ν
B
functions z/(1 z2).
−
Acknowledgement
The authorthanks theDeanship ofScientific Research atKing Faisal
University for funding this work under project number 150244.
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DepartmentofMathematicsandStatistics,CollageofScience,King
Faisal University, Al-Hasa 31982, Hofuf, Saudi Arabia.
E-mail address: [email protected]