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Aspects of univalent holomorphic functions involving multiplier transformation and Ruscheweyh derivative

Abstract

Making use multiplier transformation and Ruscheweyh derivative,we introduce a new class of analytic functions RI(γ,λ,l,α,β) defined on the open unit disc, and investigate its various characteristics. Further we obtain distortion bounds, extreme points and radii of close-to-convexity, starlikeness and convexity and neighborhood property for functions belonging to the class RI(γ,λ,l,α,β).

MSC:30C45, 30A20, 34A40.

1 Introduction

Let denote the class of functions of the form f(z)=z+ j = 2 a j z j , which are analytic and univalent in the open unit disc U={z:zC:|z|<1}. is a subclass of consisting of the functions of the form f(z)=z j = 2 | a j | z j . For functions f,gA given by f(z)=z+ j = 2 a j z j , g(z)=z+ j = 2 b j z j , we define the Hadamard product (or convolution) of f and g by (fg)(z)=z+ j = 2 a j b j z j , zU.

Definition 1.1 (Ruscheweyh [1])

For fA, nN, the operator R n is defined by R n :AA,

R 0 f ( z ) = f ( z ) , R 1 f ( z ) = z f ( z ) , , ( n + 1 ) R n + 1 f ( z ) = z ( R n f ( z ) ) + n R n f ( z ) , z U .

Remark 1.1 If fA, f(z)=z+ j = 2 a j z j , then R n f(z)=z+ j = 2 ( n + j 1 ) ! n ! ( j 1 ) ! a j z j , zU.

If fT, f(z)=z j = 2 a j z j , then R n f(z)=z j = t + 1 ( n + j 1 ) ! n ! ( j 1 ) ! a j z j , zU.

Definition 1.2 [2, 3]

For fA, nN{0}, λ,l0, the operator I(n,λ,l)f(z) is defined by the following infinite series:

I(n,λ,l)f(z):=z+ j = 2 ( λ ( j 1 ) + l + 1 l + 1 ) n a j z j .

Remark 1.2 [4]

It follows from the above definition that

I ( 0 , λ , l ) f ( z ) = f ( z ) , ( l + 1 ) I ( n + 1 , λ , l ) f ( z ) = ( l + 1 λ ) I ( n , λ , l ) f ( z ) + λ z ( I ( n , λ , l ) f ( z ) ) , z U .

Remark 1.3 The operator I(n,λ,0)= D λ n is the generalized Sălăgean operator introduced by Al-Oboudi [5], and I(n,1,0)= S n is the Sălăgean differential operator [6].

Definition 1.3 [7, 8]

Let γ,λ,l0, nN. Denote by R I n , λ , l γ the operator given by R I n , λ , l γ :AA,

R I n , λ , l γ f(z)=(1γ) R n f(z)+γI(n,λ,l)f(z),zU.

Remark 1.4 If fA, f(z)=z+ j = 2 a j z j , then

R I n , λ , l γ f(z)=z+ j = 2 { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j ,zU.

If fT, f(z)=z j = 2 a j z j , then

R I n , λ , l γ f(z)=z j = 2 { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j ,zU.

Remark 1.5 The operator R I n , λ , 0 γ f(z)=R D λ , γ n f(z) which was introduced in [9] and the operator R I n , 1 , 0 γ f(z)= L γ n f(z) which was introduced in [10].

Following the work of Najafzadeh and Pezeshki [11] we can define the class RI(γ,λ,l,α,β) as follows.

Definition 1.4 For γ,λ,l0, 0α<1 and 0<β1, let RI(γ,λ,l,α,β) be the subclass of consisting of functions that satisfying the inequality

| R I n , λ , l μ , γ f ( z ) 1 2 ν ( R I n , λ , l μ , γ f ( z ) α ) ( R I n , λ , l μ , γ f ( z ) 1 ) |<β,
(1.1)

where

R I n , λ , l μ , γ f(z)=(1μ) R I n , λ , l γ f ( z ) z +μ ( R I n , λ , l γ f ( z ) ) ,
(1.2)

0<ν1.

Remark 1.6 If fT, f(z)=z j = 2 a j z j , then

R I n , λ , l μ , γ f ( z ) = 1 j = t + 1 [ 1 + μ ( j 1 ) ] × { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j 1 , z U .

Remark 1.7 The class RI(γ,λ,0,α,β)=RD(γ,λ,α,β) defined and studied in [12] and RI(γ,1,0,α,β)=L(γ,α,β) defined and studied in [13].

2 Coefficient bounds

In this section we obtain coefficient bounds and extreme points for functions in RI(γ,λ,l,α,β).

Theorem 2.1 Let the function fT. Then fRI(γ,λ,l,α,β) if and only if

j = t + 1 ( 1 + μ ( j 1 ) ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j < 2 β ν ( 1 α ) .
(2.1)

The result is sharp for the function F(z) defined by

F(z)=z 2 β ν ( 1 α ) ( 1 + μ ( j 1 ) ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } z j ,jt+1.

Proof Suppose f satisfies (2.1). Then for |z|<1, we have

| R I n , λ , l μ , γ f ( z ) 1 | β | 2 ν ( R I n , λ , l μ , γ f ( z ) α ) ( R I n , λ , l μ , γ f ( z ) 1 ) | = | j = t + 1 ( 1 + μ ( j 1 ) ) { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j 1 | β | 2 ν ( 1 α ) ( 2 ν 1 ) j = t + 1 { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } × [ 1 + μ ( j 1 ) ] a j z j 1 | j = t + 1 [ 1 + μ ( j 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a k 2 β ν ( 1 α ) + j = t + 1 β ( 2 ν 1 ) ( 1 + μ ( j 1 ) ) { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j = j = t + 1 [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j 2 β ν ( 1 α ) < 0 .

Hence, by using the maximum modulus Theorem and (1.1), fRI(γ,λ,l,α,β). Conversely, assume that

| R I n , λ , l μ , γ f ( z ) 1 2 ν ( R I n , λ , l μ , γ f ( z ) α ) ( R I n , λ , l μ , γ f ( z ) 1 ) | = | j = t + 1 [ 1 + μ ( j 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j 1 2 ν ( 1 α ) j = t + 1 [ 1 + μ ( j 1 ) ] ( 2 ν 1 ) { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j 1 | < β , z U .

Since Re(z)|z| for all zU, we have

Re { j = t + 1 [ 1 + μ ( j 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j 1 2 ν ( 1 α ) j = t + 1 [ 1 + μ ( j 1 ) ] ( 2 ν 1 ) { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } a j z j 1 } < β .
(2.2)

By choosing choose values of z on the real axis so that R I n , λ , l μ , γ f(z) is real and letting z1 through real values, we obtain the desired inequality (2.1). □

Corollary 2.2 If fT is in RI(γ,λ,l,α,β), then

a j 2 β ν ( 1 α ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } ,jt+1,
(2.3)

with equality only for functions of the form F(z).

Theorem 2.3 Let f 1 (z)=z and

f j ( z ) = z 2 β ν ( 1 α ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } z j , j t + 1 ,
(2.4)

for 0α<1, 0<β1, γ,λ,l0 and 0<ν1. Then f(z) is in the class RI(γ,λ,l,α,β) if and only if it can be expressed in the form

f(z)= j = t ω j f j (z),
(2.5)

where ω j 0 and j = 1 ω j =1.

Proof Suppose f(z) can be written as in (2.5). Then

f(z)=z j = t + 1 ω j 2 β ν ( 1 α ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } z j .

Now,

j = t + 1 [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) ω j × 2 β ν ( 1 α ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } = j = t + 1 ω j = 1 ω 1 1 .

Thus fRI(γ,λ,l,α,β).

Conversely, let fRI(γ,λ,l,α,β). Then by using (2.3), setting

ω j = [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) a j ,jt+1,

and ω 1 =1 j = 2 ω j , we have f(z)= j = t ω j f j (z). This completes the proof of Theorem 2.3. □

3 Distortion bounds

In this section we obtain distortion bounds for the class RI(γ,λ,l,α,β).

Theorem 3.1 If fRI(γ,λ,l,α,β), then

r 2 β ν ( 1 α ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } r t + 1 | f ( z ) | r + 2 β ν ( 1 α ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } r t + 1
(3.1)

holds if the sequence { σ j ( γ , λ , l , β , ν ) } j = t + 1 is non-decreasing, and

1 2 β ν ( 1 α ) ( t + 1 ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } r t | f ( z ) | 1 + 2 β ν ( 1 α ) ( t + 1 ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } r t
(3.2)

holds if the sequence { σ j ( γ , λ , l , β , ν ) j } j = t + 1 is non-decreasing, where

σ j (γ,β,ν)= [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } .

The bounds in (3.1) and (3.2) are sharp, for f(z) given by

f(z)=z 2 β ν ( 1 α ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } z t + 1 ,z=±r.
(3.3)

Proof In view of Theorem 2.1, we have

j = t + 1 a j 2 β ν ( 1 α ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } .
(3.4)

We obtain

|z||z | t + 1 j = t + 1 a j |f(z)||z|+|z | t + 1 j = t + 1 a j .

Thus

r 2 β ν ( 1 α ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } r t + 1 | f ( z ) | r + 2 β ν ( 1 α ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } r t + 1 .
(3.5)

Hence (3.1) follows from (3.5). Further,

j = t + 1 j a j 2 β ν ( 1 α ) ( 1 + μ t ) [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } .

Hence (3.2) follows from

1 r t j = t + 1 j a j | f (z)|1+ r t j = t + 1 j a j .

 □

4 Radius of starlikeness and convexity

The radii of close-to-convexity, starlikeness, and convexity for the class RI(γ,λ,l,α,β) are given in this section.

Theorem 4.1 Let the function fT belong to the class RI(γ,λ,l,α,β), Then f(z) is close-to-convex of order δ, 0δ<1, in the disc |z|<r, where

r:= inf j t + 1 [ ( 1 δ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν j ( 1 α ) ] 1 t .
(4.1)

The result is sharp, with extremal function f(z) given by (2.3).

Proof For given fT we must show that

| f (z)1|<1δ.
(4.2)

By a simple calculation we have

| f (z)1| j = t + 1 j a j |z | t .

The last expression is less than 1δ if

j = t + 1 j 1 δ a j |z | t <1.

We use the fact that fRI(γ,λ,l,α,β) if and only if

j = t + 1 [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) a j 1.

Equation (4.2) holds true if

j 1 δ |z | t j = t + 1 [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) .

Or, equivalently,

|z | t j = t + 1 ( 1 δ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν j ( 1 α ) ,

which completes the proof. □

Theorem 4.2 Let fRI(γ,λ,l,α,β). Then

  1. 1.

    f is starlike of order δ, 0δ<1, in the disc |z|< r 1 , where

    r 1 = inf j t + 1 { ( 1 δ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) ( j δ ) } 1 t .
  2. 2.

    f is convex of order δ, 0δ<1, in the disc |z|< r 2 where,

    r 2 = inf j t + 1 { ( 1 δ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν j ( j 1 ) ( 1 α ) } 1 t .

Each of these results is sharp for the extremal function f(z) given by (2.5).

Proof 1. For 0δ<1 we need to show that

| z f ( z ) f ( z ) 1|<1δ.
(4.3)

We have

| z f ( z ) f ( z ) 1|| j = t + 1 ( j 1 ) a j | z | t 1 j = t + 1 a j | z | t |.

The last expression is less than 1δ if

j = t + 1 ( j δ ) 1 δ a j |z | t <1.

We use the fact that fRI(γ,λ,l,α,β) if and only if

j = t + 1 [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) a j <1.

Equation (4.3) holds true if

j δ 1 δ |z | t < [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) .

Or, equivalently,

|z | t < ( 1 δ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) ( j δ ) ,

which yields the starlikeness of the family.

  1. 2.

    Using the fact that f is convex if and only z f is starlike, we can prove (2) with a similar way of the proof of (1). The function f is convex if and only if

    |z f (z)|<1δ.
    (4.4)

We have

| z f ( z ) | | j = t + 1 j ( j 1 ) a j | z | t 1 | < 1 δ , j = t + 1 j ( j 1 ) 1 δ a j | z | t 1 < 1 .

We use the fact that fRI(γ,λ,l,α,β) if and only if

j = t + 1 [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) a j <1.

Equation (4.4) holds true if

j ( j 1 ) 1 δ |z | t 1 < [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν ( 1 α ) ,

or, equivalently,

|z | t 1 < ( 1 δ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } 2 β ν j ( j 1 ) ( 1 α ) ,

which yields the convexity of the family. □

5 Neighborhood property

In this section we study neighborhood property for functions in the class RI(γ,λ,l,α,β).

Definition 5.1 For functions f belong to of the form and ε0, we define (ηε)-neighborhood of f by

N ε η (f)= { g ( z ) A : g ( z ) = z + j = 2 b j z j , j = 2 j η + 1 | a j b j | ε } ,

where η is a fixed positive integer.

By using the following lemmas we will investigate the (ηε)-neighborhood of function in RI(γ,λ,l,α,β).

Lemma 5.1 Let 1β<1, if g(z)=z+ j = 2 b j z j satisfies

j = 2 j ρ + 1 | b j | 2 β ν ( 1 α ) 1 + β ( 2 ν 1 )

then g(z)RI(γ,λ,l,α,β).

Proof By using of Theorem 2.1, it is sufficient to show that

[ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) ρ + ( 1 γ ) ( ρ + j 1 ) ! ρ ! ( j 1 ) ! } 2 β ν ( 1 α ) = j ρ + 1 2 β ν ( 1 α ) [ 1 + β ( 2 ν 1 ) ] .

But

[ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) ρ + ( 1 γ ) ( ρ + j 1 ) ! ρ ! ( j 1 ) ! } 2 β ν ( 1 α ) j ρ + 1 2 β ν ( 1 α ) [ 1 + β ( 2 ν 1 ) ] .

Therefore it is enough to prove that

Q(j,ρ)= γ ( 1 + λ ( j 1 ) + l l + 1 ) ρ + ( 1 γ ) ( ρ + j 1 ) ! ρ ! ( j 1 ) ! j ρ + 1 1,

the result follows because the last inequality holds for all jt+1. □

Lemma 5.2 Let f(z)=z k = 2 a k z k T, γ,λ,l0, 0α<1, 0<β1 and ε0. If f ( z ) + ϵ z 1 + ϵ RI(γ,λ,l,α,β), then

j = t + 1 j ρ + 1 a j 2 β ν ( 1 α ) ( 1 + ϵ ) ( t + 1 ) ρ + 1 [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } ,

where either ρ=0 or ρ=1. The result is sharp with the extremal function

f(z)=z 2 β ν ( 1 α ) ( 1 + ϵ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } z t + 1 ,zU.

Proof Letting g(z)= f ( z ) + ϵ z 1 + ϵ we have

g(z)=z j = t + 1 a j 1 + ϵ z j ,zU.

In view of Theorem 2.3, g(z)= j = 1 η j g j (z) where η j 0, j = 1 η j =1,

g 1 (z)=z

and

g j (z)=z 2 β ν ( 1 α ) ( 1 + ϵ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } z j ,jt+1.

So we obtain

g ( z ) = η 1 z + j = t + 1 η j [ z 2 β ν ( 1 α ) ( 1 + ϵ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } z j ] = z j = t + 1 η k 2 β ν ( 1 α ) ( 1 + ϵ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } z j .

Since η j 0 and j = 2 η j 1, it follows that

j = t + 1 a k sup j t + 1 j ρ + 1 2 β ν ( 1 α ) ( 1 + ϵ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! } .

Since whenever ρ=0 or ρ=1 we conclude that

W(j,ρ,γ,α,β,ϵ)= j ρ + 1 2 β ν ( 1 α ) ( 1 + ϵ ) [ 1 + μ ( j 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ ( j 1 ) + l l + 1 ) n + ( 1 γ ) ( n + j 1 ) ! n ! ( j 1 ) ! }

is a decreasing function of j, the result will follow. The proof is complete. □

Theorem 5.1 Let ρ=0 or ρ=1 and suppose 0β<1 and

1 θ < [ 1 + μ ( t 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } 2 β ν ( 1 α ) ( 1 + ϵ ) ( t + 1 ) η + 1 [ 1 + μ ( t 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } ,

f(z)T and f ( z ) + ϵ z 1 + ϵ RI(γ,λ,l,α,β), then the (ηε)-neighborhood of f is the subset of RI(λ,λ,l,α,β), where

ε 2 ( 1 α ) { θ γ [ 1 + μ ( t 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } β γ [ 1 + θ ( 2 ν 1 ) ] ( 1 + ϵ ) ( t + 1 ) η + 1 } / ( [ 1 + θ ( 2 ν 1 ) ] [ 1 + μ ( t 1 ) ] × [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } ) .

The result is sharp.

Proof For f(z)=z j = 2 | a j | z j , let g(z)=z+ j = 2 b j z j be in N ε η (f). So by Lemma 5.2, we have

j = 2 j η + 1 | b j | = j = 2 j η + 1 | a j b j a j | ε + 2 β ν ( 1 α ) ( 1 + ϵ ) ( t + 1 ) η + 1 [ 1 + μ ( t 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } .

By using Lemma 5.1, g(z)L(γ,α,β) if

ε+ 2 β ν ( 1 α ) ( 1 + ϵ ) ( t + 1 ) η + 1 [ 1 + μ ( t 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } 2 θ ν ( 1 α ) 1 + θ ( 2 ν 1 ) ,

that is, ε 2 ( 1 α ) { θ γ [ 1 + μ ( t 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } β γ [ 1 + θ ( 2 ν 1 ) ] ( 1 + ϵ ) ( t + 1 ) η + 1 } [ 1 + θ ( 2 ν 1 ) ] [ 1 + μ ( t 1 ) ] [ 1 + β ( 2 ν 1 ) ] { γ ( 1 + λ t + l l + 1 ) n + ( 1 γ ) ( n + t ) ! n ! t ! } , and the proof is complete. □

Author’s contributions

The author drafted the manuscript, read and approved the final manuscript.

References

  1. 1.

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Correspondence to Alina Alb Lupaş.

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Keywords

  • analytic functions
  • univalent functions
  • radii of starlikeness and convexity
  • neighborhood property
  • Salagean operator
  • Ruscheweyh operator