On the subclasses associated with the Bessel-Struve kernel functions

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by  Saiful R. Mondal

ON THE SUBCLASSES ASSOCIATED WITH THE BESSEL-STRUVE
KERNEL FUNCTIONS
∗
SAIFUL R. MONDAL , AL DHUAIN MOHAMMED
Abstract. The article investigate the necessary and sufficient conditions for the nor-
malized Bessel-struve kernel functions belonging to the classes λ(α) , λ(α). Some
T L
linear operators involving the Bessel-Struve operator are also considered.
6
1
0 1. Introduction
2

n In this article we will consider the class of all analytic functions defined in the unit
a disk D = z C : z < 1 . Suppose thatH is the subclass of consisting of function
J which are{nor∈malize|d|by f(}0) = 0 = f (0) A1 and also univalent iHn U. Thus each function
6 ′
−
2 f posses the power series
∈ A

] ∞
V f(z) = z + a zn. (1)
n
C n=2
X
h. We also consider the subclass of consisting of the function have the power series as
J H
t
a
∞
m f(z) = z b zn, b > 0. (2)
n n
−
[ n=2
X
0v1 gT(hze) H=azda+mar∞nd=2prbonzdnuc∈t H(oriscodnevfionleudtioans) of two function f(z) = z +P∞n=2anzn ∈ H and
6
1 P (f g)(z) = z + ∞ a b zn,z D.
7 ∗ n n ∈
0 n=2
X
1. Dixit and Pal [14] introduced the subclass τ(A,B) consisting the function f which
R ∈ H
0 satisfies the inequality
6
1 f (z) 1
′
: − < 1,
v (A B)τ B[f (z) 1]
i (cid:12) − − ′ − (cid:12)
X for τ C 0 and 1 B(cid:12)(cid:12)< A 1. The class τ(A(cid:12)(cid:12),B) is the generalization of many
r well k∈nown\{su}bclasse−s, f≤or ex(cid:12)ampl≤e, if τ = 1 and AR= (cid:12) B = α [0,1), then τ(A,B) is
a
− ∈ R
the subclass of H which consist the functions satisfying the inequality ff′′((zz))+−11 < α which
was studied by Padmanabhan [14], Caplinger and Causey [6] and m(cid:12)any oth(cid:12)ers. If the
(cid:12) (cid:12)
function f of the form (1) belong to the class τ(A,B), then (cid:12) (cid:12)
R
(A B) τ
a − | |, n N. (3)
n
| | ≤ n ∈
The bounds given in (3) is sharp.
2010 Mathematics Subject Classification. 30C45.
Keywordsandphrases. Bessel-struveoperator,Starlikefunction,convexfunction,Hadamardproduct
.
Corresponding authors.
∗
1
2 SAIFULR. MONDAL∗ , AL DHUAINMOHAMMED
In this article we will consider two well-known subclass of and denoted as (α) and
λ
H T
(α). The function f in the subclass (α) satisfy the analytic criteria
λ λ
L T
zf (z)+λz2f (z)
′ ′′
(α) := f : Re > α ,
λ
T ∈ A (1 λ)f(z)+λzf (z)
(cid:26) (cid:18) − ′ (cid:19) (cid:27)
while f (α) have the analytic characterization as
λ
∈ L
λz3f (z)+(1+2λ)z2f (z)+zf (z)
′′′ ′′ ′
(α) := f : Re > α .
λ
L ∈ A zf (z)+λz2f (z)
(cid:26) (cid:18) ′ ′′ (cid:19) (cid:27)
Here z D, 0 α < 1 and 0 λ < 1. For more details about this classes and it’s
∈ ≤ ≤
generalization see [1–3, 17] and references therein. The sufficient coefficient condition by
which a function f as defined in (1) belongs to the class (α) is
λ
∈ A T
∞
(nλ λ+1)(n α) a 1 α, (4)
n
− − | | ≤ −
n=2
X
and f is in (α) is
λ
L
∞
n(nλ λ+1)(n α) a 1 α. (5)
n
− − | | ≤ −
n=2
X
The significance of the class (α) and (α) is that, it will reduce to some classical
λ λ
subclass of for specific choicTe of λ. ForLexample, if λ = 0, (α) = S (α) is the class of
λ ∗
all starlike fHunctions of order α with respect to the origin anTd (α) = C(α) is the class
λ
L
of all starlike functions of order α.
Denote (α) = (α) and (α) = (α) . Then (4) and (5) are respectively
Tλ∗ Tλ ∩J L∗λ Lλ ∩J
the necessary and sufficient condition for any f is in (α) and (α).
∈ J Tλ∗ L∗λ
Bessel and Struve functions arise in many problems of applied mathematics and math-
ematical physics. The properties of these functions were studied by many researchers in
the past years from many different point of views. Recently the Bessel-Struve kernel and
the so-called Bessel-Struve intertwining operator have been the subject of some research
from the point of view of the operator theory, see [9–11] and the references therein. The
Bessel-Struve kernel function S is defined by the series
ν
Γ(ν +1)Γ n+1
S (z) = 2 zn,
ν √πn!Γ n +ν +1
n 0 2 (cid:0) (cid:1)
X≥
where ν > 1. This function is a particular c(cid:0)ase when(cid:1)λ = 1 of the unique solution
1
−
S (λx) of the initial value problem
ν
λ Γ(ν +1)
L u(z) = λ2u(z), u(0) = 1,u(0) = 1 ,
ν 1 ′ √πΓ(ν + 3)
2
where for ν > 1/2 the expression L stands for the Bessel-Struve operator defined by
ν
−
d2u 2ν +1 du du
L u(z) = (z)+ (z) (0)
ν
dz2 x dz − dz
(cid:18) (cid:19)
with an infinitely differentiable function u on R.
Motivated by the results connecting various subclasses with the Bessel functions [4, 12],
the Struve functions [13, 18], the hypergeometric function and the Confluent hypergeo-
metric functions [7, 16], in Section 2 we obtain several conditions by which the normalized
Bessel-Struve kernel functions zS is the member of (α) and (α). Also determined
ν λ λ
T L
the condition for which z(2 S (z)) (α) and z(2 S (z)) (α).
− ν ∈ Tλ∗ − ν ∈ L∗λ
ON THE SUBCLASSES ASSOCIATED WITH THE BESSEL-STRUVE KERNEL FUNCTIONS 3
2. Main Result
2.1. Inclusion properties of the Bessel-Struve functions. Forconvenience through-
out in the sequel , we use the following notations
∞
zS (z) = z + c (ν)zn, (6)
ν n 1
−
n=2
X
and
∞
Φ(z) = z(2 S (z)) = z c (ν)zn, (7)
ν n 1
− − −
n=2
X
Γ(ν+1)Γ(n+1)
where c (ν) = 2 .
n √πn!Γ(n+ν+1)
2
Further from (6) a calculation yield
∞
z2S (z)+zS (z) = z + nc (ν)zn. (8)
ν′ ν n 1
−
n=2
X
A differentiation of both side of (8) two times with respect to z, gives
∞
z3S (z)+3z2S (z)+zS (z) = z + n2c (ν)zn, (9)
ν′′ ν′ ν n 1
−
n=2
X
and
∞
z4S (z)+6z3S (z)+7z2S (z)+zS = z + n3c (ν)zn. (10)
ν′′′ ν′′ ν′ ν n 1
−
n=2
X
Now we will state and proof our main results.
Theorem 2.1. For ν > 1/2 and 0 α,λ < 1, let
− ≤
λS (1)+(1 λα)S (1)+(1 α)S (1) 2(1 α). (11)
ν′′ − ν′ − ν ≤ −
Then the normalized Bessel-Struve function zS (z) (α).
ν λ
∈ T
Proof. Consider the identity
∞
zS (z) = z + c (ν)zn.
ν n 1
−
n=2
X
Then by virtue of (4) it is enough to show that F(n,λ,α) 1 α, where
≤ −
∞
F(n,λ,α) := (nλ λ+1)(n α)c (ν)
n 1
− − −
n=2
X
The right hand side expression of F(n,λ,α) can be rewrite as
∞ ∞ ∞
F(n,λ,α) = λ n2c (ν)+(1 λ(1+α)) nc (ν)+α(λ 1) c (ν). (12)
n 1 n 1 n 1
− − − − −
n=2 n=2 n=2
X X X
4 SAIFULR. MONDAL∗ , AL DHUAINMOHAMMED
Now for z = 1, it is evident from (6)– (9) that
| |
∞
S (1) = 1+ znc (ν),
ν n 1
−
n=2
X
∞
S (1)+S (1) = 1+ nc (ν),
ν′ ν n 1
−
n=2
X
∞
S (1)+3S (1)+S (1) = 1+ n2znc (ν).
ν′′ ν′ ν n 1
−
n=2
X
Thus (12) reduce to
F(n,λ,α) = λS (1)+(1 λα+2λ)S (1)+(1 α)(S (1) 1).
ν′′ − ν′ − ν −
and is bounded above by 1 α if (11) holds. Thus the proof is completed. (cid:3)
−
Corollary 2.1. The normalized Bessel-Struve kernel function is starlike of order α
∈
[0,1) with respect to origin if S (1)+(1 α)S (1) 2(1 α).
ν′ − ν ≤ −
Theorem 2.2. For ν > 1/2 and 0 α,λ < 1, suppose that
− ≤
λS (1)+(5λ+1 λα)S (1)+(4λ 2λα α+3)S (1)+(1 α)S (1) 2(1 α).
ν′′′ − ν′′ − − ν′ − ν ≤ −
(13)
Then the normalized Bessel-Struve function zS (α).
ν λ
∈ L
Proof. By virtue of (5), it suffices to prove that G(n,λ,α) 1 α, where
≤ −
∞
G(n,λ,α) = n(nλ λ+1)(n α)c (ν)
n 1
− − −
n=2
X
∞ ∞ ∞
= λ n3c (ν)+(1 λ(1+α)) n2c (ν)+α(λ 1) nc (ν).
n 1 n 1 n 1
− − − − −
n=2 n=2 n=2
X X X
For z = 1 and using (8)-(10), it follows that
| |
G(n,λ,α) = λ(S (1)+7S (1)+6S (1)+S (1) 1)
ν′′′ ν′′ ν′ ν −
+(1 λ(1+α))(S (1)+3S (1)+S (1) 1)+α(λ 1)(S (1)+S (1) 1)
− ν′′ ν′ ν − − ν′ ν −
= λS (1)+(5λ+1 λα)S (1)+(4λ 2λα α+3)S (1)+(1 α)(S (1) 1)
ν′′′ − ν′′ − − ν′ − γ,β −
which is bounded above by 1 α if (13) holds, and hence the conclusion. (cid:3)
−
Corollary 2.2. The normalized Bessel-Struve kernel function is convex of of order α
∈
[0,1) if
S (1)+(3 α)S (1)+(1 α)S (1) 2(1 α).
ν′′ − ν′ − ν ≤ −
Remark 2.1. The condition (11) is necessary and sufficient for z(2 S (z)) (α),
ν ∗
− ∈ T
while z(2 S (z)) (α) if and only if the condition (13) hold.
ν ∗
− ∈ L
2.2. Operators involving the Bessel-Struve functions and its inclusion proper-
ties. In this section we considered the linear operator : defined by
ν
J A → A
∞
f(z) = zS (z) f(z) = z + c (ν)a zn,
ν ν n 1 n
J ∗ −
n=2
X
In the next result we will study the action of the Bessel-struve operator on the class
τ(A,B).
R
ON THE SUBCLASSES ASSOCIATED WITH THE BESSEL-STRUVE KERNEL FUNCTIONS 5
Theorem 2.3. Suppose that ν > 1 and f τ(A,B). For α,λ (0,1], if
−2 ∈ R ∈
(A B) τ λS (1)+(1 λα+2λ)S (1)+(1 α)(S (1) 1) 1 α, (14)
− | |{ ν′′ − ν′ − ν − ≤ −
then (f) (α).
ν λ
J ∈ L
Proof. Let f be of the form (1) belong to the class τ(A,B). By virtue of the inequality
R
(5) it suffices to show that G(n,λ,α) 1 α.
≤ −
It follows from (3) that the coefficient a for each f τ(A,B) satisfy the inequality
n
∈ R
a (A B) τ /n, and hence
n
| | ≤ − | |
∞
G(n,λ,α) (A B) τ (nλ λ+1)(n α)c (ν)
n 1
≤ − | | − − −
n=2
X
∞ ∞
= (A B) τ λ n2c (ν)+(1 λ(1+α)) nc (ν)
n 1 n 1
− | | − − −
n=2 n=2
X X
∞
+α(λ 1) c (ν)
n 1
− − !
n=2
X
Using (8) and (9), the right hand side of the above inequality reduce to
G(n,λ,α) = (A B) τ λ(S (1)+3S (1)+S (1) 1)
− | |{ ν′′ ν′ ν −
+(1 λ(1+α))(S (1)+S (1) 1)+α(λ 1)(S (1) 1)
− ν′ ν − − ν − }
= (A B) τ λS (1)+(2λ λα+1)S (1)+(1 α)(S (1) 1)
− | |{ ν′′ − ν′ − ν − }
It is evident from the hypothesis 14 that G(n,λ,α) is bounded above by 1 α and hence
− (cid:3)
the conclusion.
z
Theorem 2.4. For ν > 1/2, let (z) := (2 S (t))dt. Then for α,λ (0,1], the
− Qν 0 − ν ∈
operator (α) if and only if
Qν ∈ L∗λ R
λS (1)+(2λ λα+1)S (1)+(1 α)S (1) 2(1 α). (15)
ν′′ − ν′ − ν ≤ −
Proof. Note that
∞ zn
(z) = z c (ν) .
ν n 1
Q − − n
n=2
X
It is enough to show that
∞ cn 1(ν)
n(nλ λ+1)(n α) − 1 α.
− − n ≤ −
n=2 (cid:18) (cid:19)
X
Since
∞ cn 1(ν) ∞
n(nλ λ+1)(n α) − = (nλ λ+1)(n α)cn 1(ν),
− − n − − −
n=2 (cid:18) (cid:19) n=2
X X
proceeding as the proof of Theorem 2.1, we get
∞ cn 1(ν)
n(nλ−λ+1)(n−α) −n = λSν′′(1)+(2λ−λα+1)Sν′(1)+(1−α)Sν(1),
n=2 (cid:18) (cid:19)
X
which is bounded above by 1 α if and only if (15) (cid:3)
−
6 SAIFULR. MONDAL∗ , AL DHUAINMOHAMMED
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Department of Mathematics, King Faisal University, Al-Hasa 31982, Saudi Arabia,
Email: [email protected] ; [email protected]