The confluent hypergeometric function is given by the power series

where a, c are complex numbers such that c ≠ 0, −1, −2, . . . and (a)_n is the Pochhammer symbol defined in terms of the Gamma function, by

is convergent for all finite values of z, see [12].

This suggests that the series

is convergent for a, c, m > 0.

Very recently, Porwal and Kumar [11] introduced the confluent hypergeometric distribution (CHD) whose probability mass function is

It is easy to see that for a = c it reduces to the Poisson distribution.

Definition 2.1

If X is a discrete random variable which can take the values x₁, x₂, x₃, . . . with respective probabilities p₁, p₂, p₃, . . . then expectation of X, denoted by E(X), is defined as

Definition 2.2

The r^th moment of a discrete probability distribution about X = 0 is defined by

Here μ1′ and μ2′-(μ1′)2 are known as the mean and variance of the distribution.

Definition 2.3

The moment generating function (m.g.f.) of a random variable X is denoted by M_X(t) and defined by

The proof of the following theorem is straight forward so we only state the result.

Theorem 2.1

The moment generating function of the confluent hypergeometric Distribution is given by

Remark 2.1

If we put a = c in the expressions μ1′, μ2′, μ3′, μ4′ and in Theorem 2.1, then we obtain the corresponding results of Poisson distribution.

Let denote the class of functions f(z) of the form

which are analytic in the open unit disc U = {z : z ∈ C and |z| < 1} and satisfy the normalization condition f(0) = f′(0) − 1 = 0. Further, we denote by the subclass of consisting of functions of the form (3.1) which are also univalent in U.

In 1997, Bharti et al. [1] introduced the subclasses k-uniformly convex functions of order α and corresponding class of k-starlike functions of order α as follows

A function f of the form (3.1) is in k − UCV (α), if and only if, it satisfy the following condition

For α = 0 the class k − UCV(α) reduce to the class k − UCV introduced and studied by Kanas and Wisniowska [6] and for k = 1, α = 0 it reduce to the class of uniformly convex functions UCV studied by Goodman [3]. Using the Alexander transform we can obtain the class k − S_p(α) in the following way f ∈ k − UCV(α) ⇔ zf′ ∈ k − S_p(α). For more results on these directions we refer the reader to [4, 5, 7, 8, 14, 15] and references therein.

A function is said to be in the class Pγτ(β) if it satisfies the following inequality

where 0 ≤ γ < 1, β < 1, τ ∈ C/{0} and z ∈ U. The class Pγτ(β) was introduced by Swaminathan [17].

Next, we introduce the classes Sλ* and C_λ as follows

and

From (3.3) and (3.4) it is easy to see that

The classes Sλ* and C_λ were introduced by Ponnusamy and Rønning [9].

Recently, Porwal [10] introduce a power series whose coefficients are probabilities of Poisson distribution

and we note that, by ratio test the radius of convergence of above series is infinity.

The convolution (or Hadamard product) of two series f(z)=∑n=0∞anzn and g(z)=∑n=0∞bnzn is defined as the power series

Now, we introduce a new series I(a; c; m; z) whose coefficients are probabilities of confluent hypergeometric distribution

where a, c, m > 0.

Now, we consider a linear operator defined by

The Poisson distribution series is a recent topic of study in Geometric Function Theory. It established a connection between probability distribution and Geometric Function Theory. Motivated by results of [10] and on connections between the various subclasses of analytic univalent functions by using hypergeometric functions (see [2], [9]), we establish a number of connections between the classes Pγτ(β), k − UCV (α), k − S_p(α), C_λ and Sλ* by applying the convolution operator Ω(a; c; m).

To establish our main results, we shall require the following lemmas.

Lemma 4.1

([1]) A functionis in k − UCV(α), if it satisfies the following condition

Remark 4.1

It was also found that the condition (4.1) is necessary if is of the form

Lemma 4.2

([1]) A functionis in k − S_p(α) if it satisfies the following inequality

The condition (4.3) is also necessary for functions of the form (4.2).

Lemma 4.3

([6]) Letand have the form (3.1). If for some k, 0 ≤ k < ∞, the inequality

holds, then f ∈ k − UCV. The number 1/k + 2 can not be increased.

Lemma 4.4

([9]) Letbe of the form (3.1). If

thenf∈Sλ*.

Lemma 4.5

([9])

Letbe of the form (3.1). If

then f ∈ C_λ.

We further note that when f(z) is of the form (4.2), the conditions (4.4) and (4.5) are both necessary and sufficient for belonging to the classes Sλ* and C_λ, respectively.

Lemma 4.6

([17]) Iff∈Pγτ(β)is of the form (3.1) then

Theorem 4.1

If a, c, m > 0, k ≥ 0, 0 ≤ α < 1, f∈Pγτ(β), 0 < γ ≤ 1, 0 ≤ β < 1 and the inequality

is satisfied then Ω(a; c; m) f(z) ∈ k − UCV (α).

Theorem 4.2

If a, c > 1, m > 0, k ≥ 0, 0 ≤ α < 1, f∈Pγτ(β), 0 < γ ≤ 1, 0 ≤ β < 1 and the inequality

is satisfied then Ω(a; c; m) f(z) ∈ k − S_p(α).

Theorem 4.3

Let a, c > 1, m > 0, f∈Pγτ(β); 0 < γ ≤ 1, β <1, λ > 0 and the inequality

is satisfied thenΩ(a;c;m)f(z)∈Sλ*.

Theorem 4.4

Let a, c, m > 0, f∈Pγτ(β); 0 < γ ≤ 1, β <1 and λ > 0 and the inequality

is satisfied then Ω(a; c; m)f(z) ∈ C_λ.

Theorem 4.5

Let a, c, m > 0, k ≥ 0, 0 ≤ α < 1 and the inequality

is satisfied then Ω(a; c; m) f(z) maps f(z) ∈ S of the form (3.1) into k − UCV (α).

Theorem 4.6

Let a, c, m > 0, k ≥ 0, 0 ≤ α < 1 and the inequality

is satisfied, then Ω(a; c; m) f(z) maps f(z) ∈ S of the form (3.1) into k − S_p(α).

Theorem 4.7

Let a, c, m > 0, λ > 0 and the inequality

is satisfied then Ω(a; c; m) f(z) maps f(z) ∈ S of the form (3.1) into C_λ.

Theorem 4.8

Let a, c, m > 0, λ > 0 and the inequality

is satisfied then Ω(a; c; m) f(z) mapsof the form (3.1) intoSλ*.

In the following theorem, we obtain analogues results in connection with a particular integral operator G(a, c, m, z) which is defined as follows

Theorem 5.1

Let f be of the form (3.1) is in the classPγτ(β)with a, c, m > 0 and the inequality (4.8) is satisfied then G(a, c, m, z) defined by (5.1) is in the class k − UCV (α).

Theorem 5.2

Let f be defined by (3.1) in the classwith a, c, m > 0 and the inequality (4.10) is satisfied then G(a, c, m, z) defined by (5.1) is in k − UCV (α).

Theorem 5.3

Let f be defined by (3.1) in the classwith a, c, m > 0 and the inequality

is satisfied then G(a, c, m, z) is in the class k − S_p(α).

Theorem 5.4

Let f be defined by (3.1) in the classwith a, c, m > 0 and the inequality

is satisfied then G(a, c, m, z) is in the class S^*(λ).

Theorem 5.5

Let f be defined by (3.1) in the classwith a, c, m > 0 and the inequality (4.11) is satisfied then G(a, c, m, z) is in the class C(λ).

Remark 5.1

If we put a = c in Theorems 4.1–5.5, then we obtain the corresponding results of Srivastava and Porwal [16].

The author is thankful to the referee for his/her valuable comments and suggestions.