On the subclasses associated with the Bessel-Struve kernel functions

ON THE SUBCLASSES ASSOCIATED WITH THE BESSEL-STRUVE KERNEL FUNCTIONS

arXiv:1601.07160v1 [math.CV] 26 Jan 2016

SAIFUL R. MONDAL∗ , AL DHUAIN MOHAMMED

Abstract. The article investigate the necessary and sufficient conditions for the normalized Bessel-struve kernel functions belonging to the classes Tλ (α) , Lλ (α). Some linear operators involving the Bessel-Struve operator are also considered.

1. Introduction In this article we will consider the class H of all analytic functions defined in the unit disk D = {z ∈ C : |z| < 1}. Suppose that A is the subclass of H consisting of function which are normalized by f (0) = 0 = f ′ (0) − 1 and also univalent in U. Thus each function f ∈ A posses the power series ∞ X an z n . (1) f (z) = z + n=2

We also consider the subclass J of H consisting of the function have the power series as ∞ X f (z) = z − bn z n , bn > 0. (2) n=2

The Hadamard (or convolution) of two function f (z) = z + P product n b z ∈ H is defined as g(z) = z + ∞ n=2 n (f ∗ g)(z) = z +

∞ X n=2

P∞

n=2 an z

n

∈ H and

an bn z n , z ∈ D.

Dixit and Pal [14] introduced the subclass Rτ (A, B) consisting the function f ∈ H which satisfies the inequality ′ f (z) − 1 (A − B)τ − B[f ′ (z) − 1] < 1,

for τ ∈ C \ {0} and −1 ≤ B < A ≤ 1. The class Rτ (A, B) is the generalization of many well known subclasses, for example, if τ = 1 and A = −B = α ∈ [0, 1), then Rτ (A, B) is ′ (z)−1 the subclass of H which consist the functions satisfying the inequality ff ′ (z)+1 < α which was studied by Padmanabhan [14], Caplinger and Causey [6] and many others. If the function f of the form (1) belong to the class Rτ (A, B), then |an | ≤

(A − B)|τ | , n

n ∈ N.

(3)

The bounds given in (3) is sharp. 2010 Mathematics Subject Classification. 30C45. Key words and phrases. Bessel-struve operator, Starlike function , convex function , Hadamard product . ∗ Corresponding authors.

1

SAIFUL R. MONDAL∗ , AL DHUAIN MOHAMMED

2

In this article we will consider two well-known subclass of H and denoted as Tλ (α) and Lλ (α). The function f in the subclass Tλ (α) satisfy the analytic criteria     zf ′ (z) + λz 2 f ′′ (z) Tλ (α) := f ∈ A : Re >α , (1 − λ)f (z) + λzf ′ (z)

while f ∈ Lλ (α) have the analytic characterization as     3 ′′′ λz f (z) + (1 + 2λ)z 2 f ′′ (z) + zf ′ (z) >α . Lλ (α) := f ∈ A : Re zf ′ (z) + λz 2 f ′′ (z) 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 ∈ A as defined in (1) belongs to the class Tλ (α) is ∞ X (nλ − λ + 1)(n − α)|an | ≤ 1 − α, (4) n=2

and f is in Lλ (α) is

∞ X n=2

n(nλ − λ + 1)(n − α)|an | ≤ 1 − α.

(5)

The significance of the class Tλ (α) and Lλ (α) is that, it will reduce to some classical subclass of H for specific choice of λ. For example, if λ = 0, Tλ (α) = S∗ (α) is the class of all starlike functions of order α with respect to the origin and Lλ (α) = C(α) is the class of all starlike functions of order α. Denote Tλ∗ (α) = Tλ (α) ∩ J and L∗λ (α) = Lλ (α) ∩ J . Then (4) and (5) are respectively the necessary and sufficient condition for any f ∈ J is in Tλ∗ (α) and L∗λ (α). Bessel and Struve functions arise in many problems of applied mathematics and mathematical 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 Sν is defined by the series
X Γ(ν + 1)Γ n+1 2
zn , Sν (z) = √ n + ν + 1 πn!Γ 2 n≥0 where ν > −1. This function is a particular case when λ1 = 1 of the unique solution Sν (λx) of the initial value problem Lν u(z) = λ21 u(z),

λ1 Γ(ν + 1) u(0) = 1, u′(0) = √ , πΓ(ν + 32 )

where for ν > −1/2 the expression Lν stands for the Bessel-Struve operator defined by   2ν + 1 du du d2 u (z) − (0) Lν u(z) = 2 (z) + dz x dz dz 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 hypergeometric functions [7, 16], in Section 2 we obtain several conditions by which the normalized Bessel-Struve kernel functions zSν is the member of Tλ (α) and Lλ (α). Also determined the condition for which z(2 − Sν (z)) ∈ Tλ∗ (α) and z(2 − Sν (z)) ∈ L∗λ (α).

ON THE SUBCLASSES ASSOCIATED WITH THE BESSEL-STRUVE KERNEL FUNCTIONS

3

2. Main Result 2.1. Inclusion properties of the Bessel-Struve functions. For convenience throughout in the sequel , we use the following notations zSν (z) = z +

∞ X

cn−1 (ν)z n ,

(6)

n=2

and Φ(z) = z(2 − Sν (z)) = z − where cn (ν) =

∞ X

cn−1 (ν)z n ,

(7)

n=2

Γ(ν+1)Γ( n+1 2 ) √ . πn!Γ( n +ν+1) 2

Further from (6) a calculation yield z

2

Sν′ (z)

+ zSν (z) = z +

∞ X

ncn−1 (ν)z n .

(8)

n=2

A differentiation of both side of (8) two times with respect to z, gives z 3 Sν′′ (z) + 3z 2 Sν′ (z) + zSν (z) = z +

∞ X

n2 cn−1 (ν)z n ,

(9)

n=2

and z

4

Sν′′′ (z)

+ 6z

3

Sν′′ (z)

+ 7z

2

Sν′ (z)

+ zSν = z +

∞ X

n3 cn−1 (ν)z n .

(10)

n=2

Now we will state and proof our main results. Theorem 2.1. For ν > −1/2 and 0 ≤ α, λ < 1, let λSν′′ (1) + (1 − λα)Sν′ (1) + (1 − α)Sν (1) ≤ 2(1 − α).

(11)

Then the normalized Bessel-Struve function zSν (z) ∈ Tλ (α). Proof. Consider the identity zSν (z) = z +

∞ X

cn−1 (ν)z n .

n=2

Then by virtue of (4) it is enough to show that F (n, λ, α) ≤ 1 − α, where ∞ X F (n, λ, α) := (nλ − λ + 1)(n − α)cn−1 (ν) n=2

The right hand side expression of F (n, λ, α) can be rewrite as F (n, λ, α) = λ

∞ X n=2

2

n cn−1 (ν) + (1 − λ(1 + α))

∞ X n=2

ncn−1 (ν) + α(λ − 1)

∞ X n=2

cn−1 (ν). (12)

SAIFUL R. MONDAL∗ , AL DHUAIN MOHAMMED

4

Now for |z| = 1, it is evident from (6)– (9) that Sν (1) = 1 +

∞ X

z n cn−1 (ν),

n=2

Sν′ (1) + Sν (1) = 1 + Sν′′ (1) + 3Sν′ (1) + Sν (1) = 1 +

∞ X

n=2 ∞ X

ncn−1 (ν), n2 z n cn−1 (ν).

n=2

Thus (12) reduce to

F (n, λ, α) = λSν′′ (1) + (1 − λα + 2λ)Sν′ (1) + (1 − α)(Sν (1) − 1).

and is bounded above by 1 − α if (11) holds. Thus the proof is completed.



Corollary 2.1. The normalized Bessel-Struve kernel function is starlike of order α ∈ [0, 1) with respect to origin if Sν′ (1) + (1 − α)Sν (1) ≤ 2(1 − α). Theorem 2.2. For ν > −1/2 and 0 ≤ α, λ < 1, suppose that

λSν′′′ (1) + (5λ + 1 − λα)Sν′′ (1) + (4λ − 2λα − α + 3)Sν′ (1) + (1 − α)Sν (1) ≤ 2(1 − α). (13)

Then the normalized Bessel-Struve function zSν ∈ Lλ (α). Proof. By virtue of (5), it suffices to prove that G(n, λ, α) ≤ 1 − α, where ∞ X G(n, λ, α) = n(nλ − λ + 1)(n − α)cn−1 (ν) n=2 ∞ X

=λ

n=2

3

n cn−1 (ν) + (1 − λ(1 + α))

For |z| = 1 and using (8)-(10), it follows that

∞ X n=2

2

n cn−1 (ν) + α(λ − 1)

∞ X

ncn−1 (ν).

n=2

G(n, λ, α) = λ(Sν′′′ (1) + 7Sν′′ (1) + 6Sν′ (1) + Sν (1) − 1)

+ (1 − λ(1 + α))(Sν′′(1) + 3Sν′ (1) + Sν (1) − 1) + α(λ − 1)(Sν′ (1) + Sν (1) − 1)

= λSν′′′ (1) + (5λ + 1 − λα)Sν′′(1) + (4λ − 2λα − α + 3)Sν′ (1) + (1 − α)(Sγ,β (1) − 1)

which is bounded above by 1 − α if (13) holds, and hence the conclusion.



Corollary 2.2. The normalized Bessel-Struve kernel function is convex of of order α ∈ [0, 1) if Sν′′ (1) + (3 − α)Sν′ (1) + (1 − α)Sν (1) ≤ 2(1 − α). Remark 2.1. The condition (11) is necessary and sufficient for z(2 − Sν (z)) ∈ T ∗ (α), while z(2 − Sν (z)) ∈ L∗ (α) if and only if the condition (13) hold. 2.2. Operators involving the Bessel-Struve functions and its inclusion properties. In this section we considered the linear operator Jν : A → A defined by ∞ X Jν f (z) = zSν (z) ∗ f (z) = z + cn−1 (ν)an z n , n=2

In the next result we will study the action of the Bessel-struve operator on the class Rτ (A, B).

ON THE SUBCLASSES ASSOCIATED WITH THE BESSEL-STRUVE KERNEL FUNCTIONS

5

Theorem 2.3. Suppose that ν > − 21 and f ∈ Rτ (A, B). For α, λ ∈ (0, 1], if

(A − B)|τ |{λSν′′ (1) + (1 − λα + 2λ)Sν′ (1) + (1 − α)(Sν (1) − 1) ≤ 1 − α,

(14)

then Jν (f ) ∈ Lλ (α). Proof. Let f be of the form (1) belong to the class Rτ (A, B). By virtue of the inequality (5) it suffices to show that G(n, λ, α) ≤ 1 − α. It follows from (3) that the coefficient an for each f ∈ Rτ (A, B) satisfy the inequality |an | ≤ (A − B)|τ |/n, and hence G(n, λ, α) ≤ (A − B)|τ |

∞ X n=2

= (A − B)|τ | λ

(nλ − λ + 1)(n − α)cn−1 (ν) ∞ X

+α(λ − 1)

2

n cn−1 (ν) + (1 − λ(1 + α))

n=2 ∞ X

cn−1 (ν)

n=2

!

∞ X

ncn−1 (ν)

n=2

Using (8) and (9), the right hand side of the above inequality reduce to G(n, λ, α) = (A − B)|τ |{λ(Sν′′(1) + 3Sν′ (1) + Sν (1) − 1)

+ (1 − λ(1 + α))(Sν′ (1) + Sν (1) − 1) + α(λ − 1)(Sν (1) − 1)}

= (A − B)|τ |{λSν′′(1) + (2λ − λα + 1)Sν′ (1) + (1 − α)(Sν (1) − 1)} It is evident from the hypothesis 14 that G(n, λ, α) is bounded above by 1 − α and hence the conclusion.  Rz Theorem 2.4. For ν > −1/2, let Qν (z) := 0 (2 − Sν (t))dt. Then for α, λ ∈ (0, 1], the operator Qν ∈ L∗λ (α) if and only if λSν′′ (1) + (2λ − λα + 1)Sν′ (1) + (1 − α)Sν (1) ≤ 2(1 − α).

(15)

Proof. Note that ∞ X

zn Qν (z) = z − cn−1 (ν) . n n=2 It is enough to show that ∞ X n=2

Since ∞ X n=2

n(nλ − λ + 1)(n − α)

n(nλ − λ + 1)(n − α)



cn−1 (ν) n





=

cn−1 (ν) n

∞ X n=2



≤ 1 − α.

(nλ − λ + 1)(n − α)cn−1 (ν),

proceeding as the proof of Theorem 2.1, we get   ∞ X cn−1 (ν) n(nλ − λ + 1)(n − α) = λSν′′ (1) + (2λ − λα + 1)Sν′ (1) + (1 − α)Sν (1), n n=2

which is bounded above by 1 − α if and only if (15)



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SAIFUL R. MONDAL∗ , AL DHUAIN MOHAMMED

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