Open Access

( φ , α , δ , λ , Ω ) p -Neighborhood for some classes of multivalent functions

Journal of Inequalities and Applications20132013:152

https://doi.org/10.1186/1029-242X-2013-152

Received: 13 December 2012

Accepted: 21 March 2013

Published: 3 April 2013

Abstract

In the present paper, we obtain some interesting results for neighborhoods of multivalent functions. Furthermore, we give an application of Miller and Mocanu’s lemma.

MSC:30C45.

Keywords

neighborhood multivalent function Miller and Mocanu’s lemma

1 Introduction and definitions

Let A denote the class of functions f of the form
f ( z ) = z + n = 2 a n z n
which are analytic in the open unit disk
U = { z : z C  and  | z | < 1 } .
We denote by A ( p , n ) the class of functions f of the form
f ( z ) = z p + k = n a k + p z k + p ( n , p N = { 1 , 2 , } )

which are analytic and multivalent in the open unit disk U.

The concept of neighborhood for f A was first given by Goodman [1]. The concept of δ-neighborhoods N δ ( f ) of analytic functions f A was first introduced by Ruscheweyh [2]. Walker [3] defined a neighborhood of analytic functions having positive real part. Owa et al. [4] generalized of the results given by Walker. In 1996, Altıntaş and Owa [5] gave ( n , δ ) -neighborhoods for functions f A with negative coefficients. In 2007, new definitions for neighborhoods of analytic functions f A were considered by Orhan et al. [6]. The authors gave the following definition of neighborhoods:

For f , g A , f is said to be ( α , δ ) -neighborhood for g if it satisfies
| f ( z ) e i α g ( z ) | < δ ( z U )

for some π α π and δ > 2 ( 1 cos α ) . They denote this neighborhood by ( α , δ ) N ( g ) .

Also, they saw that f ( α , δ ) M ( g ) if it satisfies
| f ( z ) z e i α g ( z ) z | < δ ( z U )

for some π α π and δ > 2 ( 1 cos α ) .

In 2009, Altuntaş et al. [7] gave the following definition for neighborhood of analytic functions f A ( p , n ) .

For f , g A ( p , n ) , f is said to be ( α , δ ) p -neighborhood for g if it satisfies
| f ( z ) z p 1 e i α g ( z ) z p 1 | < δ ( z U )

for some π α π and δ > p 2 ( 1 cos α ) . They denote this neighborhood by ( α , δ ) p N ( g ) .

Also, they saw that f ( α , δ ) p M ( g ) if it satisfies
| f ( z ) z p e i α g ( z ) z p | < δ ( z U )

for some π α π and δ > 2 ( 1 cos α ) .

Recently, Frasin [8] introduced the following definition of ( α , β , δ ) -neighborhood for analytic function f in the form
f ( z ) = z n = 2 a n z n ( a n 0 ) .
(1.1)
Let f be defined by (1.1). Then f is said to be ( α , β , δ ) -neighborhood for g = z n = 2 b n z n ( b n 0 ) if it satisfies
| e i α ( D k f ( z ) ) e i β ( D k g ( z ) ) | < δ

for some π α , β π and δ > 2 ( 1 cos ( α β ) ) .

The differential operator D k was introduced by Salagean [9].

Now, we give the following equalities for the functions f A ( p , n )
We define : A ( p , n ) A ( p , n ) such that
( f ( z ) ) = ( 1 p Ω λ ) D Ω f ( z ) + λ p z ( D Ω f ( z ) ) ( 0 λ 1 p Ω , Ω N { 0 } ) .
(1.2)

We denote by ( Ω , λ ) the class of analytic functions of the form (1.2) in U.

For f , g ( Ω , λ ) , f is said to be ( φ , α , δ , λ , Ω ) p -neighborhood for g if it satisfies
| e i φ ( f ( z ) ) z p 1 e i α ( g ( z ) ) z p 1 | < δ ( z U )

for some π φ α π and δ > p 2 ( 1 cos ( φ α ) ) . We denote this neighborhood by ( φ , α , δ , λ , Ω ) p N ( g ) .

Also, we say that f ( φ , α , δ , λ , Ω ) p M ( g ) if it satisfies
| e i φ ( f ( z ) ) z p e i α ( g ( z ) ) z p | < δ ( z U )

for some π φ α π and δ > 2 ( 1 cos ( φ α ) ) .

We discuss some properties of f belonging to ( φ , α , δ , λ , Ω ) p N ( g ) and ( φ , α , δ , λ , Ω ) p M ( g ) .

2 Main results

Theorem 2.1 If f ( Ω , λ ) satisfies
(2.1)

for some π φ α π and δ > p 2 ( 1 cos ( φ α ) ) , then f ( φ , α , δ , λ , Ω ) p N ( g ) .

Proof By virtue of (1.2), we can write
If
k = n ( k + p p ) Ω ( k + p ) ( 1 + λ k p Ω 1 ) | a k + p e i α b k + p | δ p 2 { 1 cos ( φ α ) } ,
then we see that
| e i φ ( f ( z ) ) z p 1 e i α ( g ( z ) ) z p 1 | < δ ( z U ) .

Thus, f ( φ , α , δ , λ , Ω ) p N ( g ) . □

Example 2.2

For given
g ( z ) = z p + k = n B k + p ( φ , α , δ , λ , Ω ) z k + p ( Ω , λ ) ( n , p N = { 1 , 2 , } )
we consider
f ( z ) = z p + k = n A k + p ( φ , α , δ , λ , Ω ) z k + p ( Ω , λ ) ( n , p N = { 1 , 2 , } )
with
A k + p = p Ω ( δ p 2 ( 1 cos ( φ α ) ) ) ( k + p ) Ω + 2 ( 1 + λ k p Ω 1 ) ( k + p 1 ) ( n + p 1 ) e i φ + e i ( α φ ) B k + p .
Then we have that
(2.2)
Finally, in view of the telescopic sum, we can write
k = n 1 ( k + p ) ( k + p 1 ) = lim q k = n q { 1 ( k + p 1 ) 1 ( k + p ) } = lim q { 1 ( n + p 1 ) 1 ( p + q ) } = 1 n + p 1 .
(2.3)
Using (2.3) in (2.2), we have
k = n ( k + p p ) Ω ( k + p ) ( 1 + λ k p Ω 1 ) | e i φ A k + p e i α B k + p | = ( δ p 2 ( 1 cos ( φ α ) ) ) .

Therefore, f ( φ , α , δ , λ , Ω ) p N ( g ) .

Corollary 2.3 If f ( Ω , λ ) satisfies
k = n ( k + p p ) Ω ( k + p ) ( 1 + λ k p Ω 1 ) | | a k + p | | b k + p | | δ p 2 ( 1 cos ( φ α ) )

for some π φ α π , δ > p 2 { 1 cos ( φ α ) } , and arg ( a k + p ) arg ( b k + p ) = α φ ( n , p N = { 1 , 2 , } ), then f ( φ , α , δ , λ , Ω ) p N ( g ) .

Proof By Theorem 2.1, we see the inequality (2.1) which implies that f ( φ , α , δ , λ , Ω ) p N ( g ) .

Since arg ( a k + p ) arg ( b k + p ) = α φ , if arg ( a k + p ) = φ k + p , we see arg ( b k + p ) = φ k + p α + φ . Therefore,
e i φ a k + p e i α b k + p = e i φ | a k + p | e i φ k + p e i α | b k + p | e i ( φ k + p α + φ ) = ( | a k + p | | b k + p | ) e i ( φ k + p + φ )
implies that
| e i φ a k + p e i α b k + p | = | | a k + p | | b k + p | | .
(2.4)

Using (2.4) in (2.1), the proof of the corollary is complete. □

Theorem 2.4 If f ( Ω , λ ) satisfies
k = n ( k + p p ) Ω ( 1 + λ k p Ω 1 ) | e i φ a k + p e i α b k + p | δ 2 ( 1 cos ( α φ ) )

for some π φ α π and δ > 2 { 1 cos ( φ α ) } , then f ( φ , α , δ , λ , Ω ) p M ( g ) .

The proof of this theorem is similar with Theorem 2.1.

Corollary 2.5 If f ( Ω , λ ) satisfies
k = n ( k + p p ) Ω ( 1 + λ k p Ω 1 ) | | a k + p | | b k + p | | δ 2 ( 1 cos ( φ α ) )

for some π φ α π , δ > 2 { 1 cos ( φ α ) } , and arg ( a k + p ) arg ( b k + p ) = α φ , then f ( φ , α , δ , λ , Ω ) p M ( g ) .

Next, we derive the following theorem.

Theorem 2.6 If f ( φ , α , δ , λ , Ω ) p N ( g ) , 0 φ < α π and arg ( e i φ a k + p e i α b k + p ) = k φ , then
k = n ( k + p p ) Ω ( k + p ) ( 1 + λ k p Ω 1 ) | e i φ a k + p e i α b k + p | δ p { cos φ cos α } .
Proof For f ( φ , α , δ , λ , Ω ) p N ( g ) , we have
Let us consider z such that arg z = φ . Then z k = | z | k e i k φ . For such a point z U , we see that
This implies that
{ k = n ( k + p p ) Ω ( k + p ) ( 1 + λ k p Ω 1 ) | e i φ a k + p e i α b k + p | | z | k + p ( cos φ cos α ) } 2 < δ 2
or
p ( cos φ cos α ) + k = n ( k + p p ) Ω ( k + p ) ( 1 + λ k p Ω 1 ) | e i φ a k + p e i α b k + p | | z | k < δ
for z U . Letting | z | 1 , we have that
k = n ( k + p p ) Ω ( k + p ) ( 1 + λ k p Ω 1 ) | e i φ a k + p e i α b k + p | δ p ( cos φ cos α ) .

 □

Theorem 2.7 f ( φ , α , δ , λ , Ω ) p M ( g ) , 0 φ < α π and arg ( e i φ a k + p e i α b k + p ) = k φ , then
k = n ( k + p p ) Ω ( 1 + λ k p Ω 1 ) | e i φ a k + p e i α b k + p | δ + cos α cos φ .

The proof of this theorem is similar with Theorem 2.6.

Remark 2.8 Taking φ = 0 , Ω = 0 , λ = 0 and p = 1 in Theorem 2.6, we obtain the following theorem due to Orhan et al. [6].

Theorem 2.9 If f ( α , δ ) N ( g ) and arg ( a n e i α b n ) = ( n 1 ) φ ( n = 2 , 3 , 4 , ), then
n = 2 n | a n e i α b n | δ + cos α 1 .

Remark 2.10 Taking φ = 0 , Ω = 0 and λ = 0 in Theorem 2.6, we obtain the following theorem due to Altuntaş et al. [7].

Theorem 2.11 If f ( α , δ ) p N ( g ) and arg ( a k + p e i α b k + p ) = k φ , then
k = n ( k + p ) | a k + p e i α b k + p | δ p ( 1 cos α ) .

We give an application of following lemma due to Miller and Mocanu [10].

Lemma 2.12 Let the function
w ( z ) = b n z n + b n + 1 z n + 1 + b n + 2 z n + 2 + ( n N )

be regular in the unit disk U with w ( z ) 0 ( z U ). If z 0 = r 0 e i θ 0 ( r 0 < 1 ) and | w ( z 0 ) | = max | z | r 0 | w ( z ) | , then z 0 w ( z 0 ) = m w ( z 0 ) where m is real and m n 1 .

Theorem 2.13 If f ( Ω , λ ) satisfies
| e i φ ( f ( z ) ) z p 1 e i α ( g ( z ) ) z p 1 | < δ ( p + n ) p 2 ( 1 cos ( φ α ) )
for some π φ α π and δ > ( p p + n ) 2 ( 1 cos ( φ α ) ) , then
| e i φ ( f ( z ) ) z p e i α ( g ( z ) ) z p | < δ + 2 ( 1 cos ( φ α ) ) ( z U ) .
Proof Let us define w ( z ) by
e i φ ( f ( z ) ) z p e i α ( g ( z ) ) z p = e i φ e i α + δ w ( z ) .
(2.5)
Then w ( z ) is analytic in U and w ( 0 ) = 0 . By logarithmic differentiation, we obtain from (2.5) that
e i φ ( f ( z ) ) e i α ( g ( z ) ) e i φ ( f ( z ) ) e i α ( g ( z ) ) p z = δ w ( z ) e i φ e i α + δ w ( z ) .
Since
e i φ ( f ( z ) ) e i α ( g ( z ) ) z p ( e i φ e i α + δ w ( z ) ) = p z + δ w ( z ) e i φ e i α + δ w ( z ) ,
we see that
e i φ ( f ( z ) ) z p 1 e i α ( g ( z ) ) z p 1 = p ( e i φ e i α ) + δ w ( z ) ( p + z w ( z ) w ( z ) ) .
This implies that
| e i φ ( f ( z ) ) z p 1 e i α ( g ( z ) ) z p 1 | = | p ( e i φ e i α ) + δ w ( z ) ( p + z w ( z ) w ( z ) ) | .
We claim that
| e i φ ( f ( z ) ) z p 1 e i α ( g ( z ) ) z p 1 | < δ ( p + n ) p 2 ( 1 cos ( φ α ) )

in U.

Otherwise, there exists a point z 0 U such that z 0 w ( z 0 ) = m w ( z 0 ) (by Miller and Mocanu’s lemma) where w ( z 0 ) = e i θ and m n 1 .

Therefore, we obtain that
| e i φ ( f ( z 0 ) ) z 0 p 1 e i α ( g ( z 0 ) ) z 0 p 1 | = | p ( e i φ e i α ) + δ e i θ ( p + m ) | δ ( p + m ) | p ( e i φ e i α ) | δ ( p + n ) p 2 ( 1 cos ( φ α ) ) .

This contradicts our condition in Theorem 2.13. □

Hence, there is no z 0 U such that | w ( z 0 ) | = 1 . This implies that | w ( z ) | < 1 for all z U . Thus, we have that
| e i φ ( f ( z ) ) z p e i α ( g ( z ) ) z p | = | ( e i φ e i α ) + δ w ( z ) | | e i φ e i α | + δ | w ( z ) | < δ + 2 ( 1 cos ( φ α ) ) .

Letting φ = 0 , Ω = 0 , λ = 0 and α = π 2 in Theorem 2.13, we can obtain the following corollary.

Corollary 2.14 If f A ( p , n ) satisfies
| f ( z ) z p 1 i g ( z ) z p 1 | < δ ( p + n ) p 2 ( z U )
for some δ > 2 ( p p + n ) , then
| f ( z ) z p i g ( z ) z p | < δ + 2 ( z U ) .

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Atatürk University
(2)
Department of Mathematics, Faculty of Science and Arts, Avrasya University

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© Sağsöz and Kamali; licensee Springer. 2013

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