Open Access

Aspects of univalent holomorphic functions involving multiplier transformation and Ruscheweyh derivative

Advances in Difference Equations20142014:117

https://doi.org/10.1186/1687-1847-2014-117

Received: 1 March 2014

Accepted: 9 April 2014

Published: 6 May 2014

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.

Keywords

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

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 : z C : | 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 , g A 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 ( f g ) ( z ) = z + j = 2 a j b j z j , z U .

Definition 1.1 (Ruscheweyh [1])

For f A , n N , the operator R n is defined by R n : A A ,
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 f A , 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 , z U .

If f T , 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 , z U .

Definition 1.2 [2, 3]

For f A , n N { 0 } , λ , l 0 , 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 , l 0 , n N . Denote by R I n , λ , l γ the operator given by R I n , λ , l γ : A A ,
R I n , λ , l γ f ( z ) = ( 1 γ ) R n f ( z ) + γ I ( n , λ , l ) f ( z ) , z U .
Remark 1.4 If f A , 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 , z U .
If f T , 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 , z U .

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 , l 0 , 0 α < 1 and 0 < β 1 , 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 ) = ( 1 μ ) R I n , λ , l γ f ( z ) z + μ ( R I n , λ , l γ f ( z ) ) ,
(1.2)

0 < ν 1 .

Remark 1.6 If f T , 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 f T . Then f RI ( 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 , j t + 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), f RI ( 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 z U , 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 z 1 through real values, we obtain the desired inequality (2.1). □

Corollary 2.2 If f T 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 ) ! } , j t + 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 0 α < 1 , 0 < β 1 , γ , λ , l 0 and 0 < ν 1 . 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 f RI ( γ , λ , l , α , β ) .

Conversely, let f RI ( 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 , j t + 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 f RI ( 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 f T belong to the class RI ( γ , λ , l , α , β ) , Then f ( z ) is close-to-convex of order δ, 0 δ < 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 f T we must show that
| f ( z ) 1 | < 1 δ .
(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 1 δ if
j = t + 1 j 1 δ a j | z | t < 1 .
We use the fact that f RI ( 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 f RI ( γ , λ , l , α , β ) . Then
  1. 1.
    f is starlike of order δ, 0 δ < 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, 0 δ < 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 0 δ < 1 we need to show that
| z f ( z ) f ( z ) 1 | < 1 δ .
(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 1 δ if
j = t + 1 ( j δ ) 1 δ a j | z | t < 1 .
We use the fact that f RI ( 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 ) | < 1 δ .
    (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 f RI ( 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 ε 0 , we define ( η ε ) -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 ( η ε ) -neighborhood of function in RI ( γ , λ , l , α , β ) .

Lemma 5.1 Let 1 β < 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 j t + 1 . □

Lemma 5.2 Let f ( z ) = z k = 2 a k z k T , γ , λ , l 0 , 0 α < 1 , 0 < β 1 and ε 0 . 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 , z U .
Proof Letting g ( z ) = f ( z ) + ϵ z 1 + ϵ we have
g ( z ) = z j = t + 1 a j 1 + ϵ z j , z U .
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 , j t + 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 0 β < 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 ( η ε ) -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.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, University of Oradea

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© Alb Lupaş licensee Springer. 2014

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