Article Search
eISSN 0454-8124
pISSN 1225-6951

### Articles

Kyungpook Mathematical Journal 2018; 58(3): 495-505

Published online September 30, 2018 https://doi.org/10.5666/KMJ.2018.58.3.495

### Confluent Hypergeometric Distribution and Its Applications on Certain Classes of Univalent Functions of Conic Regions

Saurabh Porwal

Department of Mathematics, UIET, CSJM University, Kanpur-208024, (U. P.), India
e-mail : saurabhjcb@rediffmail.com

Received: May 27, 2016; Revised: July 16, 2018; Accepted: August 6, 2018

The purpose of the present paper is to investigate Confluent hypergeometric distribution. We obtain some basic properties of this distribution. It is worthy to note that the Poisson distribution is a particular case of this distribution. Finally, we give a nice application of this distribution on certain classes of univalent functions of the conic regions.

Keywords: confluent hypergeometric series, univalent functions, starlike functions, convex functions, uniformly convex functions.

The confluent hypergeometric function is given by the power series

F(a;c;z)=n=0(a)n(c)n(1)nzn,

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

(a)n=Γ(a+n)Γ(a)={1,if n=0a(a+1)(a+n-1),if nN={1,2,3,}

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

This suggests that the series

F(a;c;m)=n=0(a)n(c)n(1)nmn

is convergent for a, c, m > 0.

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

P(n)=(a)nmn(c)nn!F(a;c;m),a,c,m>0,n=0,1,2,....

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

E(X)=k=1pkxk.

### Definition 2.2

The rth moment of a discrete probability distribution about X = 0 is defined by

μr=E(Xr).

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

• μ1=n=0nP(n)=n=0n(a)nmn(c)nn!F(a;c;m)=macF(a+1;c+1;m)F(a;c;m).

Similarly

• μ2=1F(a;c;m)[(a)2(c)2m2F(a+2;c+2;m)+acmF(a+1;c+1;m)].

• μ3=1F(a;c;m)[(a)3(c)3m3F(a+3;c+3;m)+3(a)2(c)2m2F(a+2;c+2;m)+acmF(a+1;c+1;m)].

• μ4=1F(a;c;m)[(a)4(c)4m4F(a+4;c+4;m)+6(a)3(c)3m3F(a+3;c+3;m)+7(a)2(c)2m2F(a+2;c+2;m)+acmF(a+1;c+1;m)].

### Definition 2.3

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

MX(t)=E(etX).

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

MX(t)=F(a;c;met)F(a;c;m).

### 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.

### 3. Application of Confluent Hypergeometric Distribution on Certain Classes of Univalent Functions of Conic Regions

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

f(z)=z+n=2anzn

which are analytic in the open unit disc U = {z : zC 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 kUCV (α), if and only if, it satisfy the following condition

{1+zf(z)f(z)}k|zf(z)f(z)|+α,         0k<,0α<1.

For α = 0 the class kUCV(α) reduce to the class kUCV 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 kSp(α) in the following way fkUCV(α) ⇔ zf′ ∈ kSp(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

|(1-γ)f(z)z+γf(z)-12τ(1-β)+(1-γ)f(z)z+γf(z)-1|<1,

where 0 ≤ γ < 1, β < 1, τC/{0} and zU. The class Pγτ(β) was introduced by Swaminathan [17].

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

Sλ*={fA:|zf(z)f(z)-1|<λ,(zU,λ>0)}

and

Cλ={fA:|zf(z)f(z)|<λ,(zU,λ>0)}.

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

f(z)Cλzf(z)Sλ*,(λ>0).

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

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

K(m,z)=z+n=2mn-1(n-1)!e-mzn,         (zU),

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=0anzn and g(z)=n=0bnzn is defined as the power series

(f*g)(z)=n=0anbnzn.

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

I(a;c;m;z)=z+n=2(a)n-1mn-1(c)n-1(n-1)!F(a;c;m)zn,

where a, c, m > 0.

Now, we consider a linear operator defined by

Ω(a;c;m)f=I(a;c;m;z)*f(z)=z+n=2(a)n-1mn-1(c)n-1(n-1)!F(a;c;m)anzn.

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), kUCV (α), kSp(α), Cλ 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 kUCV(α), if it satisfies the following condition

n=2n[n(1+k)-(k+α)]an1-α.

### Remark 4.1

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

f(z)=z-n=2anzn,an0.

### Lemma 4.2

([1]) A functionis in kSp(α) if it satisfies the following inequality

n=2[n(1+k)-(k+α)]an1-α.

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

n=2n(n-1)an1(k+2),

holds, then fkUCV. The number 1/k + 2 can not be increased.

### Lemma 4.4

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

n=2(λ+n-1)anλ,(λ>0),

thenfSλ*.

### Lemma 4.5

([9])

Letbe of the form (3.1). If

n=2n(λ+n-1)anλ,(λ>0),

then fCλ.

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 Cλ, respectively.

### Lemma 4.6

([17]) IffPγτ(β)is of the form (3.1) then

an2τ(1-β)1+γ(n-1).

### Theorem 4.1

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

(k+1)acmF(a+1;c+1;m)+(1-α)(F(a;c;m)-1)γF(a;c;m)(1-α)2τ(1-β)

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

Proof

Since

Ω(a;c;m)f(z)=z+n=2(a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)anzn.

To prove that Ω(a; c; m) f(z) ∈ kUCV (α), from Lemma 4.1, it is sufficient to show that

n=2n[n(1+k)-(k+α)]an1-α,

where

An=(a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)an,         n2.

Now, by using Lemma 4.6 and 1 + γ (n − 1) ≥ γn, we have

n=2n[n(1+k)-(k+α)](a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)an2τ(1-β)n=2n[n(1+k)-(k+α)](a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)11+γ(n-1),2τ(1-β)γF(a;c;m)n=2[n(1+k)-(k+α)](a)n-1(c)n-1mn-1(n-1)!,=2τ(1-β)γF(a;c;m)[(k+1)n=2(a)n-1(c)n-1mn-1(n-2)!+(1-α)n=2(a)n-1(c)n-1mn-1(n-1)!]=2τ(1-β)γF(a;c;m)[(k+1)acmF(a+1;c+1;m)+(1-α)(F(a;c;m)-1)].

The last expression is bounded above by 1 − α, if (4.6) holds.

This completes the proof of Theorem 4.1.

### Theorem 4.2

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

(k+1)(F(a;c;m)-1)-(k+α)m(c-1)(a-1)(F(a-1;c-1;m)-1-(a-1)(c-1)m)γ(1-α)F(α;c;m)2τ(1-β)

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

Proof

The proof of this theorem is much akin to that of Theorem 4.1 so we omit the details involved.

### Theorem 4.3

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

2τ(1-β)γF(a;c;m)[(F(a;c;m)-1)+(λ-1)m(c-1)(a-1)(F(a-1;c-1;m)-1-(a-1)(c-1)m)]λ

is satisfied thenΩ(a;c;m)f(z)Sλ*.

Proof

To prove that Ω(a;c;m)f(z)Sλ*, from Lemma 4.4 it is sufficient to prove that

n=2(n+λ-1)anλ

where

An=(a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)an,         n2.

Since fPγτ(β) using Lemma 4.6 and 1 + γ(n − 1) ≥ γn we need only to show that

n=2(n+λ-1)(a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)2τ(1-β)1+γ(n-1)λ.

Now adopting the same technique of Theorem 4.1 and performing simple calculations we obtain the required result.

### Theorem 4.4

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

2τ(1-β)F(a;c;m)γ[acmF(a+1;c+1;m)+λ(F(a;c;m)-1)]λ

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

Proof

The proof is similar to that of Theorem 4.3 therefore we omit the details.

### Theorem 4.5

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

(1+k)(a)3(c)3m3F(a+3;c+3;m)+(6+5k-α)(a)2(c)2m2F(a+2;c+2;m)+(7+4k-3α)acmF(a+1;c+1;m)1-α

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

Proof

Let be of the form (3.1). In view of Lemma 4.1 it is enough to show that

T=n=2n[n(k+1)-(k+α)](a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)an1-α.

Now

T=n=2n[n(k+1)-(k+α)](a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)ann=2n2[n(k+1)-(k+α)](a)n-1(c)n-1mn-1(n-1)!1F(a;c;m)=1F(a;c;m)[(1+k)(a)3(c)3m3F(a+3;c+3;m)+(6+5k-α)(a)2(c)2m2F(a+2;c+2;m)+(7+4k-3α)acmF(a+1;c+1;m)+(1-α)(F(a;c;m)-1)]

The last expression is bounded above by 1 − α, if (4.9) holds. Thus the proof of Theorem 4.5 is established.

### Theorem 4.6

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

(1+k)(a)2(c)2m2F(a+2;c+2;m)+(3+2k-α)acmF(a+1;c+1;m)1-α,

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

Proof

The proof of this theorem is much akin to that of Theorem 4.5. Therefore we omit the details involved.

### Theorem 4.7

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

(a)3(c)3m3F(a+3;c+3;m)+(λ+5)(a)2(c)2m2F(a+2;c+2;m)+(3λ+4)acmF(a;c;m)λ

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

Proof

The proof of this theorem is similar to that of Theorem 4.3. Therefore we omit the details involved.

### Theorem 4.8

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

(a)2(c)2m2F(a+2;c+2;m)+(λ+2)acmF(a+1;c+1;m)λF(a;c;m)

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

Proof

The proof of this theorem is similar to that of Theorem 4.3. Therefore we omit the details involved.

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

G(a,c,m,z)=0zΩ(a;c;m)f(t)tdt.

### 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 kUCV (α).

Proof

Since

G(a,c,m,z)=z+n=2(a)n-1(c)n-1mn-1n!1F(a;c;n)anzn.

To prove G(a, c, m, z) ∈ kUCV (α), adopting the technique of Theorem 4.1 and performing simple calculatios we obtain the required result.

The proof of following Theorems 5.2–5.5 are similar to Theorem 5.1 therefore we only state the results of these theorems.

### 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 kUCV (α).

### Theorem 5.3

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

(k+1)acmF(a+1;c+1;m)1-α

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

### Theorem 5.4

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

acmF(a+1;c+1;m)λ

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.

1. Bharati, R, Parvatham, R, and Swaminathan, A (1997). On subclasses of uniformly convex functions and corresponding class of starlike funcitons. Tamkang J Math. 28, 17-32.
2. Gangadharan, A, Shanmugam, TN, and Srivastava, HM (2002). Generalized hypergeometric function associated with k-uniformly convex functions. Comput Math Appl. 44, 1515-1526.
3. Goodman, AW (1991). On uniformly convex functions. Ann Polon Math. 56, 87-92.
4. Goodman, AW (1991). On uniformly starlike functions. J Math Anal Appl. 155, 364-370.
5. Kanas, S, and Srivastava, HM (2000). Linear operators associated with k-uniformly convex functions. Integral Transform Spec Funct. 9, 121-132.
6. Kanas, S, and Wisniowska, A (1999). Conic regions and k-uniform convexity. J Comput Appl Math. 105, 327-336.
7. Kanas, S, and Wisniowska, A (2000). Conic domains and starlike functions. Rev Roumaine Math Pures Appl. 45, 647-657.
8. Ma, W, and Minda, D (1992). Uniformly convex functions. Ann Polon Math. 57, 165-175.
9. Ponnusamy, S, and Rønning, F (1998). Starlikeness properties for convolutions involving hypergeometric series. Ann Univ Mariae Curie-Sklodawska Sect A. 52, 141-155.
10. Porwal, S (2014). An application of a Poisson distribution series on certain analytic functions. J Complex Anal, 3.
11. Porwal, S, and Kumar, S (2017). Confluent hypergeometric distribution and its applications on certain classes of univalent functions. Afr Mat. 28, 1-8.
12. Rainville, ED (1960). Special functions. New York: The Macmillan Co
13. Robertson, MS (1936). On the theory of univalent functions. Ann Math. 37, 374-408.
14. Rønning, F (1993). Uniformly convex functions and a corresponding class of starlike functions. Proc Amer Math Soc. 118, 189-196.
15. Srivastava, HM, and Mishra, AK (2000). Applications of fractional calculus to parabolic starlike and uniformly convex functions. Comput Math Appl. 39, 57-69.
16. Srivastava, D, and Porwal, S (2015). Some sufficient conditions for Poisson distribution series associated with conic regions. Int J Advanced Technology in Enginering Sci. 3, 229-236.
17. Swaminathan, A (2004). Certain sufficiency conditions on Gaussian hypergeometric functions. J Inequal Pure Appl Math. 5, 10.