Application of Hypergeometric Distribution Series on Certain Subclass of Analytic Functions

ISSN 2291-8639

Volume 12, Number 2 (2016), 157-162 http://www.etamaths.com

APPLICATION OF HYPERGEOMETRIC DISTRIBUTION SERIES ON CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS

TRAILOKYA PANIGRAHI∗

Abstract. The object of the present paper is to give some characterizations for hypergeometric distribution series to be in various subclasses of analytic functions.

1. Introduction

Let A denote the family of all functions f analytic in U := {z ∈ C : |z| < 1} with the usual

normalization conditionf(0) =f0(0)−1 = 0. Thusf has the following Taylor-Maclaurin series:

(1) f(z) =z+

∞

X

l=2 alzl.

LetS be the subclass ofAconsisting of all functions f of the form (1) which are univalent in U. A

functionf ∈ Ais said to be ink− U CV, the class ofk-uniformly convex function (0≤k <∞) iff ∈ S along with the property that for every circular arcγ contained in_Uwith centerξ where|ξ|< k, the image curvef(γ) is a convex arc. It is well-known that [5]f ∈k− U CV if and only if the image of the functionp, wherep(z) = 1 +zf_f000₍(_zz₎) (z∈U) is a subset of the conic region

(2) Ωk={w=u+iv:u2> k2(u−1)2+k2v2,0≤k <∞}.

The classk− ST consisting ofk-uniformly starlike functions is defined viak− U CV by the Alexander transform i.e.

f ∈k− ST ⇐⇒g∈k− U CV whereg(z) =

Z z

0 f(t)

t dt.

The classk− ST and its properties were investigated in [6]. The analytic characterization ofk− U CV andk− ST are given as below:

(3) k− U CV={f ∈ A:<

1 + zf 00_(z)

f0_(z)

> k

zf00(z) f0_(z)

(z∈U)}

and

(4) k− ST ={f ∈ A:<

_zf0_(z)

f(z)

> k

zf0(z) f(z) −1

(z∈_U)}

Note that fork= 0 andk= 1, we get 0− U CV=K, 0− ST =S∗_{, 1− U CV =U CVand 1− ST =SP,} whereK, S∗_{, U CV, SP are respectively the familiar classes of univalent convex functions, univalent} starlike functions [3], uniformly convex functions [4] (also, see [7, 12]) and parabolic starlike functions [12].

For two analytic functionsf andgin_U, the functionf is said to be subordinate tog org is said to be superordinate tof, if there exists a functionwanalytic in_Uwith|w| ≤ |z|such thatf(z) =g(w(z)). In such case, we writef ≺gorf(z)≺g(z). If the functiongis univalent in_U, thenf ≺g⇐⇒f(0) =g(0) andf(_U)⊂g(_U) (see, for detail [8]).

2010Mathematics Subject Classification. 30C45, 30C50.

Key words and phrases. analytic functions; k-uniformly convex functions; k-uniformly starlike functions;

hypergeo-metric distribution series.

c

2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

Making use of subordination between analytic functions, Aouf [1] introduced and studied the class Rλ_{(A, B, α) as follows:}

Definition 1.(see[1, with p=1]) For−1≤A < B≤1, |λ|< π₂ and 0≤α <1, we say that a function f(z)∈ Ais in the classRλ_{(A, B, α) if it satisfies the following subordination condition:}

(5) eiλf0(z)≺cosλ(1−α)1 +Az 1 +Bz +α

+isinλ.

The subordination (5) is equivalent to the inequality (6) given below:

(6)

eiλ_(f0_(z)−1)

Beiλ_f0_(z)−[Beiλ_{+ (A−B)(1−α)cosλ]}

<1 (z∈U).

For particular values of parametersA, B, αand λ, we obtain various subclasses of analytic functions studied by different researchers (for details, see [2]).

In 1998, Ponnusamy and Ronning [10] introduced and studied the classes S∗

β and Cβ consisting of

functions of the form (1) satisfying the following conditions:

(7) Sβ∗=

f ∈ A:

zf0_(z) f(z) −1

< β (z∈U, β >0) ,

and

(8) Cβ=

f ∈ A:

zf00(z) f0_(z)

< β (z∈_U, β >0) .

It is worthy to mention here that

f ∈ Cβ⇐⇒zf0 ∈ Sβ∗ (β >0).

Recently, we introduced a new seriesH(M, N, n;z) whose coefficient are probabilities of hypergeometric distribution as follows:

(9) H(M, N, n;z) =z+ _N1

n

∞

X

l=2

_M

l−1

_N−M

n−l+ 1

zl.

Let us define the linear operatorJ(M, N, n) :A −→ Agiven by

(10) J(M, N, n)f(z) =H(M, N, n;z)? f(z) =z+ _N1

n

∞

X

l=2

_M

l−1

_N−M

n−l+ 1

alzl (z∈U),

where?denote the convolution or Hadamard product between two analytic functions.

Motivated by the works of [9, 10, 13], in this paper we investigate some characterization for hyperge-ometric distribution series to be in the subclassesS∗

β and Cβ of analytic functions.

2. _{Preliminaries lemmas}

To prove our main results, we need the following lemmas.

Lemma 1. (see [1], Theorem 4 with p=1) A sufficient condition forf(z) defined by (1) to be in the classRλ(A, B, α) is

(11)

∞

X

l=2

l(1 +|B|)|al| ≤(B−A)(1−α)cosλ.

Lemma 2. (see [6]) Letf(z)∈ A. If for somek, the following inequality

(12)

∞

X

l=2

(l+k(l−1))|al| ≤1

holds true, thenf ∈k− ST.

Lemma 3. (see [5, 11]) A functionf ∈ Aof the form (1) is ink− U CV if it satisfies the condition

(13)

∞

X

l=2

Another sufficient condition for the classk− U CV is given in [7] as follows:

Lemma 4. (see [7, 11]) Letf ∈ S be of the form (1). If for somek(0≤k <∞), the inequality

(14)

∞

X

l=2

l(l−1)|al| ≤

1 k+ 2,

holds true, thenf ∈k− U CV. The number _k+21 cannot be increased. Lemma 5. (see [11]) Letf ∈ Abe of the form (1). If the inequality

(15)

∞

X

l=2

[β+l−1]|al| ≤β (β >0),

is satisfied, thenf ∈ S∗

β.

Lemma 6. (see [11]) Letf ∈ Abe of the form (1). If

(16)

∞

X

l=2

l[β+l−1]|al| ≤β (β >0),

thenf ∈Cβ.

Lemma 7. (see [1], Theorem 1 with p=1) Let the function f(z) defined by (1) be in the class Rλ_{(A, B, α), then}

(17) |al| ≤

(B−A)(1−α)cosλ

l (l≥2).

3. _{Main Results}

Unless otherwise stated, we assume throughout the sequel that−1≤A < B≤1,|λ|<π₂,0≤α <1. Theorem 1. Letk≥0. If the inequality

(18) _N1

n

M(k+ 1)A1−

_N−M

n

≤ secλ

(B−A)(1−α)−1,

where

(19) A1=

∞

X

l=2

M −1 l−2

N−M n−l+ 1

is satisfied, thenJ(M, N, n) maps the classRλ_{(A, B, α) into k− U CV.}

Proof. Let the functionf given by (1) be a member ofRλ_{(A, B, α). By (10), we have}

J(M, N, n)f(z) =z+ _N1

n

∞

X

l=2

_M

l−1

_N−M

n−l+ 1

alzl.

In view of Lemma 3, it is sufficient to show that

1

N n

∞

X

l=2

l[l(k+ 1)−k]

_M

l−1

_N−M

n−l+ 1

|al| ≤1.

By making use of Lemma 7, it is again sufficient to prove that

(20) P1=

1

N n

∞

X

l=2

[l(k+ 1)−k]

_M

l−1

_N−M

n−l+ 1

≤ secλ

Now

P1= 1

N n

∞

X

l=2

[(l−1)(k+ 1) + 1]

_M

l−1

_N−M

n−l+ 1

= _N1

n

"∞

X

l=2

(k+ 1) M!

(l−2)!(M−l+ 1)!

_N−M

n−l+ 1

+ ∞

X

l=2

_M

l−1

_N−M

n−l+ 1

#

= M(k_N+ 1)

n

∞

X

l=2

_M−1

l−2

_N−M

n−l+ 1

+ _N1

n

"_∞ X

l=0

_M

l

_N−M

n−l

−

_N−M

n

#

=M(k_N+ 1)

n

A1− N−M

n

N n

+ 1,

whereA1 is defined as in (19).

Thus, in view of (20), if the inequality (18) is satisfied, thenJ(M, N, n)(f)∈k− U CV as asserted.

The proof of Theorem 1 is complete.

Theorem 2. If the inequality

(21) M_N

n A1≤

secλ

(k+ 2)(B−A(1−α)

is satisfied, thenJ(M, N, n) maps the classRλ_{(A, B, α) into k− U CV.}

Proof. Let the function f given by (1) be a member of Rλ_{(A, B, α). By virtue of Lemma 4, it is}

sufficient to show that 1

N n

∞

X

l=2 l(l−1)

M l−1

N−M n−l+ 1

|al| ≤

1 k+ 2

Using the coefficient estimate (17), it is again sufficient to show that

(22) P2=

1

N n

∞

X

l=2 (l−1)

_M

l−1

_N−M

n−l+ 1

≤ secλ

(k+ 2)(B−A)(1−α).

Now,

P2= M

N n

∞

X

l=2

_M−1

l−2

_N−M

n−l+ 1

= M_N

n A1.

In view of (22), if the condition (21) is satisfied, thenJ(M, N, n)(f)∈k− U CVas asserted. This ends

the proof of Theorem 2.

Theorem 3. If the inequality

(23) (1 +k)−(1 +_Nk)

n

_N−M

n

− _N k

n

(M+ 1)B1≤

secλ (B−A)(1−α),

where

(24) B1=

∞

X

l=2

_{M + 1}

l

_N−M

n−l+ 1

is satisfied, thenJ(M, N, n) maps the classRλ_{(A, B, α) into k− ST.}

Proof. Let the function f given by (1) be a member of Rλ_{(A, B, α). By virtue of Lemma 2, it is}

sufficient to show that

1

N n

∞

X

l=2

[l+k(l−1)]

_M

l−1

_N−M

n−l+ 1

|al| ≤1.

Using the coefficient estimate (17), it is again sufficient to show that

(25) P3=

1

N n

∞

X

l=2

[l+k(l−1)] l

_M

l−1

_N−M

n−l+ 1

≤ secλ

Now,

P3= 1

N n

∞

X

l=2

1 + (1−1 l)k

M l−1

_N−M

n−l+ 1

= _N1

n

∞

X

l=2

(1 +k)−k l

M l−1

_N−M

n−l+ 1

= (1 +k)

"

1−

N−M n

N n

#

− k

(M + 1) N_nB1.

Therefore, in view of (25), if the inequality (23) is satisfied, thenJ(M, N, n)(f)∈k− ST as asserted.

This complete the proof of Theorem 3.

Theorem 4. Iff ∈ Rλ_{(A, B, α) and the inequality}

(26) 1−

N−M n

N n

≤

1 1 +|B|,

is satisfied, thenJ(M, N, n)(f)∈ Rλ_{(A, B, α).}

Proof. Let the functionf ∈ Agiven by (1) be a member ofRλ_{(A, B, α). By virtue of Lemma 1 and}

the coefficient inequality (17) it is sufficient to show that

(27) P4=

1

N n

∞

X

l=2

_M

l−1

_N−M

n−l+ 1

≤ 1

1 +|B|.

NowP4 is equivalently written as

P4= ∞

X

l=1

M l

N−M n−l

N n

= 1− N−M

n

N n

Thus, in view of (27), if the inequality (26) is satisfied, thenJ(M, N, n)(f)∈ Rλ_{(A, B, α). The proof}

of Theorem 4 is complete.

Theorem 5. Letβ >0,f ∈ Rλ_{(A, B, α) and the inequality}

(28) β−1

(M + 1) N_nB1− N−M

n

N n

≤

βsecλ

(B−A)(1−α)−1,

is satisfied, thenJ(M, N, n)(f)∈ S∗

β .

Proof. By making use of Lemma 5, it is sufficient to show that

∞

X

l=2

(β+l−1)

M l−1

N−M n−l+1

N n

|al| ≤β.

Sincef ∈ Rλ_{(A, B, α), using the coefficient estimate (17), it is sufficient to show that}

(29) P5=

1

N n

∞

X

l=2

β+l−1 l

M l−1

N−M n−l+ 1

≤ βsecλ (B−A)(1−α).

Now,

P5= 1

N n

∞

X

l=2

_β

−1 l

M l−1

_N

−M n−l+ 1

+ _N1

n

∞

X

l=2

_M

l−1

_N

−M n−l+ 1

= β−1

(M+ 1) N_nB1− N−M

n

N n

+ 1.

Thus, in view of (29), if the inequality (28) is satisfied, thenJ(M, N, n)(f) ∈ S∗

β as asserted. This

Theorem 6. Letβ >0. If the inequality

(30) _N1

n

M A1−β

_N−M

n

≤β

_secλ

(B−A)(1−α)−1

is satisfied, thenJ(M, N, n) maps the classRλ_{(A, B, α) into C} β.

Proof. In view of Lemma 6, it is sufficient to show that

1

N n

∞

X

l=2

l[β+l−1]

_M

l−1

_N−M

n−l+ 1

|al| ≤β.

Using coefficient inequality (17), it is enough to show that

(31) P6=

1

N n

∞

X

l=2

[β+l−1]

_M

l−1

_N−M

n−l+ 1

≤ βsecλ (B−A)(1−α).

Now the expressionP4 can be equivalently written as

P6 = β

N n

∞

X

l=2

_M

l−1

_N−M

n−l+ 1

+ ∞

X

l=2

M!

(l−2)!(M −l+ 1)!

_N−M

n−l+ 1

= β−β

N−n n

N n

+

M

N n

∞

X

l=2

_{M −1}

l−2

_N−M

n−l+ 1

= M_N

n A1−

β N−_nM N n

+β.

Thus, in view of (31) if the inequality (30) is satisfied, thenJ(M, N, n)(f)∈ Cβas desired. The proof

of Theorem 6 is thus completed.

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Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar-751024, Orissa, India

∗