Open Access

Properties for certain subclasses of meromorphic functions defined by a multiplier transformation

Journal of Inequalities and Applications20132013:46

https://doi.org/10.1186/1029-242X-2013-46

Received: 27 September 2012

Accepted: 26 January 2013

Published: 1 March 2013

Abstract

Some inclusion and convolution properties of certain subclasses of meromorphic functions associated with a family of multiplier transformations, which are defined by means of the Hadamard product (or convolution), are investigated. We also obtain closure properties for certain integral operators.

MSC:30C45, 30C80.

Keywords

meromorphic functionsubordinationstarlike of order αconvex of order αprestarlike of order αconvolutionintegral operator

1 Introduction

Let A denote the class of analytic functions f in the open unit disk U = { z C : | z | < 1 } with the usual normalization f ( 0 ) = f ( 0 ) 1 = 0 . Let S ( α ) and K ( α ) denote the subclasses of A consisting of starlike and convex functions of order α ( 0 α < 1 ) and let S ( 0 ) = S and K ( 0 ) = K . If f and g are analytic in U , we say that f is subordinate to g in U , written as f g or f ( z ) g ( z ) , if there exists a Schwarz function w such that f ( z ) = g ( w ( z ) ) ( z U ).

A function f A is said to be prestarlike of order α in U if
z ( 1 z ) 2 ( 1 α ) f ( z ) S ( α ) ( 0 α < 1 ) ,

where f g denotes the familiar Hadamard product (or convolution) of two analytic functions f and g in U . We denote this class by R ( α ) (see, for details, [1]). We note that R ( 0 ) = K and R ( 1 / 2 ) = S ( 1 / 2 ) .

Let N be the class of all functions h which are analytic and univalent in U and for which h ( U ) is convex with h ( 0 ) = 1 .

Let denote the class of functions of the form
f ( z ) = 1 z + k = 0 a k z k

which are analytic in the punctured open unit disk D = U { 0 } .

For any n N 0 = { 0 , 1 , 2 , } , we denote the multiplier transformations D λ n of functions f M by
D λ n f ( z ) = 1 z + k = 0 ( k + 1 + l λ ) n a k z k ( λ > 0 ; z D ) .
Obviously, we have
D λ s ( D λ t f ( z ) ) = D λ s + t f ( z )

for all nonnegative integers s and t. The operators D λ n and D 1 n are the multiplier transformations introduced and studied by Sarangi and Uraligaddi [2] and Uralegaddi and Somanatha [3, 4], respectively. Analogous to D λ n , we here define a new multiplier transformation I λ , μ n as follows.

Let f n ( z ) = 1 / z + k = 0 ( ( k + 1 + λ ) / λ ) n z k , n N 0 , and let f n , μ be such that
f n ( z ) f n , μ ( z ) = 1 z + k = 0 ( μ ) k + 1 ( 1 ) k + 1 z k ( μ > 0 ; z D ) ,
where ( ν ) k is the Pochhammer symbol (or the shifted factorial) defined (in terms of the gamma function) by
( ν ) k : = Γ ( ν + k ) Γ ( ν ) = { 1 if  k = 0  and  ν C { 0 } , ν ( ν + 1 ) ( ν + k 1 ) if  k N : = { 1 , 2 , }  and  ν C .
Then
I λ , μ n f ( z ) = f n , μ ( z ) f ( z ) .
(1.1)
We note that I 1 , 2 0 f ( z ) = z f ( z ) + 2 f ( z ) and I 1 , 2 1 f ( z ) = f ( z ) . It is easily verified from (1.1) that
z ( I λ , μ n + 1 f ( z ) ) = λ I λ , μ n f ( z ) ( λ + 1 ) I λ , μ n + 1 f ( z )
(1.2)
and
z ( I λ , μ n f ( z ) ) = μ I λ , μ + 1 n f ( z ) ( μ + 1 ) I λ , μ n f ( z ) .
(1.3)

The definition (1.1) of the multiplier transformation I λ , μ n is motivated essentially by the Choi-Saigo-Srivastava operator [5] for analytic functions, which includes the Noor integral operator studied by Liu [6] (also, see [79]).

We also define the function ϕ ( a , c ; z ) by
ϕ ( a , c ; z ) : = 1 z + k = 0 ( a ) k + 1 ( c ) k + 1 z k ( z U ; a R ; c R Z 0 ; Z 0 : = { 1 , 2 , } ) .
(1.4)
By using the operator I λ , μ n , we introduce the following class of analytic functions for γ > 0 , λ > 0 , s R , μ > 0 and h N :
M λ , μ n ( γ ; h ) : = { f M : ( 1 γ ) z I λ , μ n f ( z ) + γ z 2 ( I λ , μ n f ( z ) ) h ( z ) } .

In the present paper, we derive some inclusion relations, convolution properties and integral preserving properties for the class M λ , μ n ( γ ; h ) .

The following lemmas will be required in our investigation.

Lemma 1.1 [[10], Lemma 2, p.192]

Let g be analytic in U and h be analytic and convex univalent in U with h ( 0 ) = g ( 0 ) . If
g ( z ) + 1 γ z g ( z ) h ( z ) ( Re { γ } 0 ; γ 0 ) ,
(1.5)
then
g ( z ) h ˜ ( z ) = γ z γ 0 z t γ 1 h ( t ) d t h ( z )

and h ˜ is the best dominant of (1.5).

Lemma 1.2 [[1], Theorem 2.4, p.54]

Let f S ( α ) and g R ( α ) . Then for any analytic function F in U ,
g ( f F ) g f ( U ) co ¯ ( F ( U ) ) ,

where co ¯ ( F ( U ) ) denotes the convex hull of F ( U ) .

Lemma 1.3 [[11], Lemma 5, p.656]

Let 0 < a c . Then
Re { z ϕ ( a , c ; z ) } > 1 2 ( z U ) ,

where ϕ is given by (1.4).

2 Inclusion relations

Theorem 2.1 If 0 γ 1 < γ 2 , then
M λ , μ n ( γ 2 ; h ) M λ , μ n ( γ 1 ; h ) .

Proof

Let
g ( z ) = z I λ , μ n f ( z ) ( f M λ , μ n ( γ 2 ; h ) : z U ) .
(2.1)
Then the function g is analytic in U with g ( 0 ) = 1 . Differentiating both sides of (2.1), we have
( 1 + γ 2 ) z I λ , μ n f ( z ) + γ 2 z 2 ( I λ , μ n f ( z ) ) = g ( z ) + γ 2 z g ( z ) h ( z ) .
(2.2)
Hence an application of Lemma 1.1 with μ = 1 / γ 2 yields
g ( z ) h ( z ) .
(2.3)
Since 0 γ 1 / γ 2 < 1 and h is convex univalent in U, it follows from (2.1), (2.2) and (2.3) that
( 1 + γ 1 ) z I λ , μ n f ( z ) + γ 1 z 2 ( I λ , μ n f ( z ) ) = γ 1 γ 2 [ ( 1 γ 2 ) z I λ , μ n f ( z ) + γ 2 z 2 ( I λ , μ n f ( z ) ) ] + ( 1 γ 1 γ 2 ) g ( z ) h ( z ) .

Therefore f M λ , μ n ( γ 1 ; h ) , and so we complete the proof of Theorem 2.1. □

Theorem 2.2 If 0 < μ 1 μ 2 , then
M λ , μ 2 n ( γ ; h ) M λ , μ 1 n ( γ ; h ) .
Proof Let f M λ , μ 2 n ( γ ; h ) . Then
(2.4)
In view of Lemma 1.3, we see that the function z ϕ ( μ 1 , μ 2 ; z ) has the Herglotz representation
z ϕ ( μ 1 , μ 2 ; z ) = | x | = 1 d μ ( x ) 1 x z ( z U ) ,
(2.5)
where μ ( x ) is a probability measure defined on the unit circle | x | < 1 and
| x | = 1 d μ ( x ) = 1 .
Since h is convex univalent in U , it follows from (2.4) and (2.5) that
( 1 + γ ) z I λ , μ 1 n f ( z ) + γ z 2 ( I λ , μ 1 n f ( z ) ) = | x | = 1 h ( x z ) d μ ( x ) h ( z ) ,

which completes the proof of Theorem 2.2. □

Theorem 2.3 If μ > 0 , then
M λ , μ + 1 n ( γ ; h ˜ ) M λ , μ n ( γ ; h ) ,
where
h ˜ ( z ) = μ z μ 0 z t μ 1 h ( t ) d t h ( z ) .

Proof

Let
g ( z ) = ( 1 + γ ) z I λ , μ n f ( z ) + γ z 2 ( I λ , μ n f ( z ) ) ( f M ; z U ) .
(2.6)
Then from (1.4) and (2.6), we have
z 1 g ( z ) = γ μ I λ , μ + 1 n f ( z ) + ( 1 γ μ ) I λ , μ n f ( z ) .
(2.7)
Differentiating both sides of (2.6) and using (1.4), we obtain
(2.8)
By a simple calculation with (2.7) and (2.8), we get
g ( z ) + z g ( z ) μ = ( 1 + γ ) I λ , μ + 1 n f ( z ) z + γ ( I λ , μ + 1 n f ( z ) ) .
(2.9)
If f M λ , μ + 1 n ( γ ; h ) , then it follows from (2.9) that
g ( z ) + z g ( z ) μ h ( z ) ( μ > 0 ) .
Hence an application of Lemma 1.1 yields
g ( z ) h ˜ ( z ) = μ z μ 0 z t μ 1 h ( t ) d t h ( z ) ,
which shows that
f M λ , μ + 1 n ( γ ; h ˜ ) M λ , μ n ( γ ; h ) .

 □

Theorem 2.4 If s R and λ > 0 , then
M λ , μ n ( γ ; h ˜ ) M λ , μ n + 1 ( γ ; h ) ,
where
h ˜ ( z ) = λ z λ 0 z t λ 1 h ( t ) d t h ( z ) .

Proof By using the same techniques as in the proof of Theorem 2.3 and (1.5), we have Theorem 2.4 and so we omit the detailed proof involved. □

Theorem 2.5 Let γ > 0 , β > 0 and f M λ , μ n ( γ ; β h + 1 β ) . If β β 0 , where
β 0 = 1 2 ( 1 1 γ 0 1 u 1 γ 1 1 + u d u ) 1 ,
(2.10)
then f M λ , μ n ( 0 ; h ) . The bound β 0 is sharp for the function
h ( z ) = 1 1 z ( z U ) .

Proof

Let
g ( z ) = z I λ , μ n f ( z ) ( f M λ , μ n ( γ ; β h + 1 β ) ; γ > 0 ; β > 0 ) .
(2.11)
Then we have
g ( z ) + γ z g ( z ) = ( 1 + γ ) z I λ , μ n f ( z ) + γ z 2 ( I λ , μ n f ( z ) ) β h ( z ) + 1 β .
Hence an application of Lemma 1.1 yields
g ( z ) β γ z 1 γ 0 z t 1 γ 1 h ( t ) d t + 1 β = ( h ψ ) ( z ) ,
(2.12)
where
ψ ( z ) = β γ z 1 γ 0 z t 1 γ 1 1 t d t + 1 β .
(2.13)
If 0 < β β 0 , where β 0 is given by (2.10), then from (2.13), we have
Re { ψ ( z ) } = β γ 0 1 u 1 γ 1 Re { 1 1 u z d u } + 1 β > β γ 0 1 u 1 γ 1 1 + u d u + 1 β 1 2 .
By using the Herglotz representation for ψ, it follows from (2.11) and (2.12) that
z I λ , μ n f ( z ) ( h ψ ) ( z ) h ( z )

since h is convex univalent in U . This shows that f M λ , μ n ( 0 ; h ) .

For h ( z ) = 1 / ( 1 z ) and f M defined by
z I λ , μ n f ( z ) = β γ z 1 γ 0 z t 1 γ 1 1 t d t + 1 β ,
it is easy to verify that
( 1 + γ ) z I λ , μ n f ( z ) + γ z 2 ( I λ , μ n f ( z ) ) = β h ( z ) + 1 β .
Thus f M λ , μ n ( γ ; β h + 1 β ) . Furthermore, for β > β 0 , we have
Re { z I λ , μ n f ( z ) } β γ 0 1 u 1 γ 1 1 + u d u + 1 β < 1 2 ( z 1 ) ,

which implies that f M λ , μ n ( 0 ; h ) . Hence the bound β 0 cannot be increased when h ( z ) = 1 / ( 1 z ) ( z U ). □

3 Convolution properties

Theorem 3.1 If f M λ , μ n ( γ ; h ) and
Re { z g ( z ) } > 1 2 ( g M ; z U ) ,
then
f g M λ , μ n ( γ ; h ) .
Proof Let f M λ , μ n ( γ ; h ) and g M . Then we have
( 1 + γ ) z I λ , μ n ( f g ) ( z ) + γ z 2 ( I λ , μ n ( f g ) ( z ) ) = z g ( z ) ψ ( z ) ,
where
ψ ( z ) = ( 1 + γ ) z I λ , μ n f ( z ) + γ z 2 ( I λ , μ n f ( z ) ) h ( z ) .

The remaining part of the proof of Theorem 3.1 is similar to that of Theorem 2.2, and so we omit the details involved. □

Corollary 3.1 Let f M λ , μ n ( γ ; h ) be given by (1.1). Then the function
σ m ( z ) = 0 1 t S m ( t z ) d t ( z U ) ,
where
S m ( z ) = 1 z + n = 1 m 1 a n z n 1 ( m N { 1 } ; z U ) ,

is also in the class M λ , μ n ( γ ; h ) .

Proof

We have
σ m ( z ) = 1 z + n = 1 m 1 a n n 1 z n + 1 = ( f g m ) ( z ) ( m N { 1 } ) ,
(3.1)
where
f ( z ) = 1 z + n = 1 a n z n 1 M λ , μ n ( γ ; h )
and
g m ( z ) = 1 z + n = 1 m 1 z n n 1 M ,
while, it is known [4] that
Re { z g m ( z ) } = Re { 1 + n = 1 m 1 z n n + 1 } > 1 2 ( m N { 1 } ; z U ) .
(3.2)

In view of (3.1) and (3.2), an application of Theorem 3.1 leads to σ m M λ , μ n ( γ ; h ) . □

Theorem 3.2 If f M λ , μ n ( γ ; h ) and
z 2 g ( z ) R ( α ) ( g M ; z U ) ,
then
( f g ) M λ , μ n ( γ ; h ) .

Proof

By using a similar method as in the proof of Theorem 3.1, we have
( 1 + γ ) z I λ , μ n ( f g ) ( z ) + γ z 2 ( I λ , μ n ( f g ) ( z ) ) = z 2 g ( z ) ( z ψ ( z ) ) z 2 g ( z ) z ( z U ) ,
(3.3)
where
ψ ( z ) = ( 1 + γ ) z I λ , μ n f ( z ) + γ z 2 ( I λ , μ n f ( z ) ) h ( z ) .

Since h is convex univalent in U , it follows from (3.3) and Lemma 1.2 that Theorem 3.2 holds true. □

If we take α = 0 and α = 1 / 2 in Theorem 3.2, we have the following corollary.

Corollary 3.2 If f M λ , μ n ( γ ; h ) and g M satisfies one of the following conditions:
  1. (i)

    z 2 g ( z ) is convex univalent in U

     
or
  1. (ii)

    z 2 g ( z ) S ( 1 2 ) ,

     

then ( f g ) M λ , μ n ( γ ; h ) .

4 Integral operators

Theorem 4.1 If f M λ , μ n ( γ ; h ) , then the function F defined by
F ( z ) = c 1 z c 0 z t c 1 f ( t ) d t ( Re { c } > 1 )
(4.1)
is in the class M λ , μ n ( γ ; h ˜ ) , where
h ˜ ( z ) = ( c 1 ) z ( c 1 ) 0 z t c h ( t ) d t h ( z ) .
Proof Let f M λ , μ n ( γ ; h ) . Then from (4.1), we obtain
( c 1 ) f ( z ) = z F ( z ) + c F ( z ) .
(4.2)
Define the function G by
z 1 G ( z ) = ( 1 + γ ) I λ , μ n F ( z ) + γ z ( I λ , μ n F ( z ) ) ( z D ) .
(4.3)
Differentiating both sides of (4.3) with respect to z, we get
z G ( z ) G ( z ) = ( 1 + γ ) z I λ , μ n ( z F ( z ) ) + γ z 2 ( I λ , μ n ( z F ( z ) ) ) .
(4.4)
Furthermore, it follows from (4.2), (4.3) and (4.4) that
(4.5)
Since f M λ , μ n ( γ ; h ) , from (4.5), we have
G ( z ) + 1 c 1 z G ( z ) h ( z ) ( Re { c } > 1 ) ,
and so an application of Lemma 1.1 yields
G ( z ) h ˜ ( z ) = c 1 z c 1 0 z t c h ( t ) d t h ( z ) .
Therefore we conclude that
F M λ , μ n ( γ ; h ˜ ) M λ , μ n ( γ ; h ) .

 □

Theorem 4.2 If f M and F are defined as in Theorem  4.1, if
( 1 α ) z I λ , μ n F ( z ) + α z I λ , μ n f ( z ) h ( z ) ( α > 0 ) ,
(4.6)
then F M λ , μ n ( 0 ; h ˜ ) , where
h ˜ ( z ) = c 1 α z α c 1 0 z t c 1 α 1 h ( t ) h ( z ) ( Re { c } > 1 ) .

Proof

Let
G ( z ) = z I λ , μ n F ( z ) ( z D ) .
(4.7)
Then G is analytic in U with G ( 0 ) = 1 and
z G ( z ) = z 2 ( I λ , μ n F ( z ) ) + G ( z ) .
(4.8)
It follows from (4.2), (4.6), (4.7) and (4.8) that
( 1 α ) z I λ , μ n F ( z ) + α z I λ , μ n f ( z ) = ( 1 α ) z I λ , μ n F ( z ) + α c 1 [ c z I λ , μ n F ( z ) + z 2 ( I λ , μ n F ( z ) ) ] = G ( z ) + α c 1 z G ( z ) h ( z ) ( Re { c } > 1 ; α > 0 ) .

Therefore, by Lemma 1.1, we conclude that Theorem 4.2 holds true as stated. □

Theorem 4.3 Let F M λ , μ n ( γ ; h ) . If the function f is defined by
F ( z ) = c 1 z c 0 z t c 1 f ( t ) d t ( c > 1 ) ,
(4.9)
then
σ f ( σ z ) M λ , μ n ( γ ; h ) ,
where
σ = σ ( c ) = 1 + ( c 1 ) 2 1 c 1 .
(4.10)
The bound σ is sharp for the function
h ( z ) = β + ( 1 β ) 1 + z 1 z ( β 1 ; z U ) .
(4.11)
Proof We note that for F M ,
F ( z ) = F ( z ) 1 z ( 1 z ) and z F ( z ) = F ( z ) ( 1 z ( 1 z ) 2 2 z 2 ( 1 z ) ) .
Then from (4.9), we have
f ( z ) = c F ( z ) + z F ( z ) c 1 = ( F g ) ( z ) ( c > 1 ; z D ) ,
(4.12)
where
g ( z ) = 1 c 1 ( ( c 2 ) 1 z ( 1 z ) + 1 z ( 1 z ) 2 ) M .
(4.13)
Next, we show that
Re { z g ( z ) } > 1 2 ( | z | < σ ) ,
(4.14)
where σ = σ ( c ) is given by (4.10). Letting
1 1 z = R e i θ ( | z | = r < 1 ; R > 0 ) ,
we see that
cos θ = 1 + R 2 ( 1 r 2 ) 2 R and R 1 1 + r .
(4.15)
Then for (4.13) and (4.15), we have
2 Re { z g ( z ) } = 2 c 1 [ ( c 2 ) R cos θ + R 2 ( 2 cos 2 θ 1 ) ] = R 2 c 1 [ c ( 1 r 2 ) + R 2 ( 1 r 2 ) 2 2 ] + 1 R 2 c 1 [ c 1 2 r ( c 1 ) r 2 ] + 1 .
This evidently gives (4.14), which is equivalent to
Re { σ z g ( σ z ) } > 1 2 ( z U ) .
(4.16)
Let F M λ , μ n ( γ ; h ) . Then, by using (4.12) and (4.16), an application of Theorem 3.1 yields
σ f ( σ z ) = F ( z ) σ g ( σ z ) M λ , μ n ( γ ; h ) .
For h given by (4.11), we consider the function F M defined by
( 1 + γ ) z I λ , μ n F ( z ) + γ z 2 ( I λ , μ n F ( z ) ) = β + ( 1 β ) 1 + z 1 z ( β 1 ; z U ) .
(4.17)
Then from (4.3), (4.5) and (4.17), we find that
( 1 + γ ) z I λ , μ n f ( z ) + γ z 2 ( I λ , μ n f ( z ) ) = β + ( 1 β ) 1 + z 1 z + z c 1 ( β + ( 1 β ) 1 + z 1 z ) = β + ( 1 β ) ( c 1 + 2 z ( c 1 ) z 2 ) ( c 1 ) ( 1 z ) 2 = β ( z = σ ) .

Therefore we conclude that the bound σ = σ ( c ) cannot be increased for each c ( c > 1 ). □

Declarations

Acknowledgements

The authors would like to express their thanks to the referees for valuable advice regarding a previous version of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Pukyong National University
(2)
Department of Statistics, Pukyong National University

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© Cho and Yoon; licensee Springer 2013

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