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Some properties of certain subclasses of analytic functions involving adifferential operator

Abstract

In the present paper, we introduce and study certain subclasses of analyticfunctions in the open unit disk U which is defined by the differentialoperator D R λ m , n . We study and investigate some inclusionproperties of these classes. Furthermore, a generalizedBernardi-Libera-Livington integral operator is shown to be preserved for theseclasses.

MSC: 30C45.

1 Introduction

Let be aclass of functions f in the open unit disk U={zC:|z|<1} normalized by f(0)= f (0)1=0. Thus each fA has a Taylor series representation

f(z)=z+ j = 2 a j z j .
(1.1)

We denote by S(ξ) the well-known subclass of consisting of allanalytic functions which are, respectively, starlike of order ξ[1, 2]

S(ξ)= { f A : Re ( z f ( z ) f ( z ) ) > ξ , z U } ,0ξ<1.

Let be a class of all functions ϕ which are analytic andunivalent in U and for which ϕ(U) is convex with ϕ(0)=1 and Reϕ(z)>0, zU.

For two functions f and g analytic in U, we say that thefunction f is subordinate to g in U and writef(z)g(z), zU, if there exists a Schwarz functionw(z) which is analytic in U withw(0)=0 and |w(z)|<1 such that f(z)=g(w(z)), zU.

Making use of the principle of subordination between analytic functions, denote byS(ξ,ϕ)[3] a subclass of the class for0ξ<1 and ϕR which are defined by

S(ξ,ϕ)= { f A : 1 1 ξ ( z f ( z ) f ( z ) ζ ) ϕ ( z ) , z U } .

Let f,gA, where f and g are defined byf(z)=z+ j = 2 a j z j and g(z)=z+ j = 2 b j z j . Then the Hadamard product (or convolution)fg of the functions f and g is definedby

(fg)(z)=z+ j = 2 a j b j z j .

Definition 1.1 (Al-Oboudi [4])

For fA, λ0 and mN, the operator D λ m is defined by D λ m :AA,

D λ 0 f ( z ) = f ( z ) , D λ 1 f ( z ) = ( 1 λ ) f ( z ) + λ z f ( z ) = D λ f ( z ) , , D λ m f ( z ) = ( 1 λ ) D λ m 1 f ( z ) + λ z ( D λ m f ( z ) ) = D λ ( D λ m 1 f ( z ) ) , z U .

Remark 1.1 If fA and f(z)=z+ j = 2 a j z j , then D λ m f(z)=z+ j = 2 [ 1 + ( j 1 ) λ ] m a j z j , zU.

Remark 1.2 For λ=1 in the above definition, we obtain theSălăgean differential operator [5].

Definition 1.2 (Ruscheweyh [6])

For fA and 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 1.3 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.

Definition 1.3 ([7])

Let λ0 and n,mN. Denote by D R λ m , n :AA the operator given by the Hadamard product of thegeneralized Sălăgean operator D λ m and the Ruscheweyh operator R n ,

D R λ m , n f(z)= ( D λ m R n ) f(z),

for any zU and each nonnegative integer m,n.

Remark 1.4 If fA and f(z)=z+ j = 2 a j z j , then D R λ m , n f(z)=z+ j = 2 [ 1 + ( j 1 ) λ ] m ( n + j 1 ) ! n ! ( j 1 ) ! a j 2 z j , zU.

Remark 1.5 The operator D R λ m , n was studied also in [810].

For λ=1, m=n, we obtain the Hadamard productS R n [11] of the Sălăgean operator S n and the Ruscheweyh derivative R n , which was studied in [12, 13].

For m=n, we obtain the Hadamard productD R λ n [14] of the generalized Sălăgean operator D λ n and the Ruscheweyh derivative R n , which was studied in [1520].

Using a simple computation, one obtains the next result.

Proposition 1.1 ([7])

Form,nNandλ0, we have

D R λ m + 1 , n f(z)=(1λ)D R λ m , n f(z)+λz ( D R λ m , n f ( z ) )
(1.2)

and

z ( D R λ m , n f ( z ) ) =(n+1)D R λ m , n + 1 f(z)nD R λ m , n f(z).
(1.3)

By using the operator D R λ m , n f(z), we define the following subclasses of analyticfunctions for 0ζ<1 and ϕR:

S λ m , n ( ξ ) = { f A : D R λ m , n f S ( ξ ) } , S λ m , n ( ξ , ϕ ) = { f A : D R λ m , n f S ( ξ , ϕ ) } .

In particular, we set

S λ m , n ( ξ , 1 + A z 1 + B z ) = S λ m , n (ξ,A,B),1<B<A1.

Next, we will investigate various inclusion relationships for the subclasses ofanalytic functions introduced above. Furthermore, we study the results of Faisalet al.[21], Darus and Faisal [3].

2 Inclusion relationship associated with the operator D R λ m , n

First, we start with the following lemmas which we need for our main results.

Lemma 2.1 ([22, 23])

Letφ(μ,v)be a complex function such thatφ:DC, DC×C, and letμ= μ 1 +i μ 2 , v= v 1 +i v 2 . Suppose thatφ(μ,v)satisfies the following conditions:

  1. 1.

    φ(μ,v) is continuous in D,

  2. 2.

    (1,0)D and Reφ(1,0)>0,

  3. 3.

    Reφ(i μ 2 , v 1 )0 for all (i μ 2 , v 1 )D such that v 1 1 2 (1+ μ 2 2 ).

Leth(z)=1+ c 1 z+ c 2 z 2 +be analytic in U, such that(h(z),z h (z))Dfor allzU. IfRe{φh(z),z h (z)}>0, zU, thenRe{h(z)}>0.

Lemma 2.2 ([24])

Let ϕ be convex univalent in U withϕ(0)=1andRe{kϕ(z)+ν}>0, k,νC. If p is analytic in U withp(0)=1, then

p(z)+ z p ( z ) k p ( z ) + ν ϕ(z),zU,

impliesp(z)ϕ(z), zU.

Theorem 2.1 LetfA, 0ξ<1, m,nN, λ>0, then

S λ m , n + 1 (ξ) S λ m , n (ξ) S λ m , n 1 (ξ).

Proof Let f S λ m , n + 1 (ξ) and suppose that

z ( D R λ m , n f ( z ) ) D R λ m , n f ( z ) =ξ+(1ξ)h(z).
(2.1)

Since from (1.3)

(n+1) D R λ m , n + 1 f ( z ) D R λ m , n f ( z ) =n+ξ+(1ξ)h(z),

we obtain

( 1 ξ ) h ( z ) = ( n + 1 ) [ ( D R λ m , n + 1 f ( z ) ) D R λ m , n f ( z ) D R λ m , n + 1 f ( z ) D R λ m , n f ( z ) ( D R λ m , n f ( z ) ) D R λ m , n f ( z ) ] , ( 1 ξ ) z h ( z ) = ( n + 1 ) D R λ m , n + 1 f ( z ) D R λ m , n f ( z ) [ z ( D R λ m , n + 1 f ( z ) ) D R λ m , n + 1 f ( z ) ξ ( 1 ξ ) h ( z ) ] , ( 1 ξ ) h ( z ) z n + ξ + ( 1 ξ ) h ( z ) = z ( D R λ m , n + 1 f ( z ) ) D R λ m , n + 1 f ( z ) ξ ( 1 ξ ) h ( z ) , z ( D R λ m , n + 1 f ( z ) ) D R λ m , n + 1 f ( z ) ξ = ( 1 ξ ) h ( z ) + ( 1 ξ ) h ( z ) z n + ξ + ( 1 ξ ) h ( z ) .

Taking h(z)=μ= μ 1 +i μ 2 and z h (z)=v= v 1 +i v 2 , we define φ(μ,v) by

φ(μ,v)=(1ξ)μ+ ( 1 ξ ) v n + ξ + ( 1 ξ ) μ

and

Re { φ ( i μ 2 , v 1 ) } = ( 1 ξ ) ( n + ξ ) v 1 ( n + ξ ) 2 + ( 1 ξ ) 2 μ 2 2 , Re { φ ( i μ 2 , v 1 ) } ( 1 ξ ) ( n + ξ ) ( 1 + μ 2 2 ) 2 [ ( n + ξ ) 2 + ( 1 ξ ) 2 μ 2 2 ] < 0 .

Clearly, φ(μ,v) satisfies the conditions of Lemma 2.1. HenceRe{h(z)}>0, zU, implies f S λ m , n (ξ). □

Remark 2.1 Using relation (1.2) and the same techniques as to prove theearlier results, we can obtain a new similar result.

Theorem 2.2 Let fA and ϕR with

Re { ϕ ( z ) } < ξ 1 + 1 λ 1 ξ .

Then

S λ m + 1 , n (ξ,ϕ) S λ m , n (ξ,ϕ) S λ m 1 , n (ξ,ϕ).

Proof Let f(z) S λ m + 1 , n (ξ,ϕ) and set

p(z)= 1 1 ξ ( z ( D R λ m , n f ( z ) ) D R λ m , n f ( z ) ξ ) ,
(2.2)

where p is analytic in U with p(0)=1.

By using (1.2) we have

z ( D R λ m , n f ( z ) ) D R λ m , n f ( z ) = 1 λ D R λ m + 1 , n f ( z ) D R λ m , n f ( z ) 1 λ λ .

Now, by using (2.2) we get

p ( z ) = 1 1 ξ ( 1 λ D R λ m + 1 , n f ( z ) D R λ m , n f ( z ) 1 λ λ ξ ) , 1 λ D R λ m + 1 , n f ( z ) D R λ m , n f ( z ) = ξ + 1 λ λ + ( 1 ξ ) p ( z ) .
(2.3)

By using (2.2) and (2.3), we obtain

z p ( z ) = 1 1 ξ 1 λ [ z ( D R λ m + 1 , n f ( z ) ) D R λ m , n f ( z ) D R λ m + 1 , n f ( z ) D R λ m , n f ( z ) z ( D R λ m , n f ( z ) ) D R λ m , n f ( z ) ] , ( 1 ξ ) z p ( z ) = 1 λ D R λ m + 1 , n f ( z ) D R λ m , n f ( z ) [ z ( D R λ m + 1 , n f ( z ) ) D R λ m + 1 , n f ( z ) z ( D R λ m , n f ( z ) ) D R λ m , n f ( z ) ] , ( 1 ξ ) z p ( z ) = [ ζ 1 + 1 λ + ( 1 ξ ) p ( z ) ] [ z ( D R λ m + 1 , n f ( z ) ) D R λ m + 1 , n f ( z ) ( 1 ξ ) p ( z ) ξ ] , ( 1 ξ ) z p ( z ) ( 1 ξ ) p ( z ) + ζ 1 + 1 λ = z ( D R λ m + 1 , n f ( z ) ) D R λ m + 1 , n f ( z ) ξ ( 1 ξ ) p ( z ) .

Hence,

1 1 ξ [ z ( D R λ m + 1 , n f ( z ) ) D R λ m + 1 , n f ( z ) ξ ] =p(z)+ z p ( z ) ( 1 ζ ) p ( z ) + ζ 1 + 1 λ .
(2.4)

Since Re{ϕ(z)}< ξ 1 + 1 λ 1 ξ implies Re{(1ξ)p(z)+ξ1+ 1 λ }>0, applying Lemma 2.2 to (2.4) we have thatf(z) S λ m , n (ξ,ϕ), as required. □

Remark 2.2 By using relation (1.3) and the same techniques as to prove theearlier results, we can obtain a new similar result.

Corollary 2.3 Let 1 + A 1 + B < ξ 1 + 1 λ 1 ξ for1<B<A1, then

S λ m + 1 , n (ξ,A,B) S λ m , n (ξ,A,B) S λ m 1 , n (ξ,A,B).

Proof Taking ϕ(z)= 1 + A z 1 + B z , 1<B<A1 in Theorem 2.2, we get thecorollary. □

3 Integral-preserving properties

In this section, we present several integral-preserving properties for the subclassesof analytic functions defined above. We recall the generalizedBernardi-Libera-Livington integral operator [25] defined by

F c [ f ( z ) ] = c + 1 z c 0 z t c 1 f(t)dt=z+ j = 2 c + 1 j + c a j z c ,fA,c>1,
(3.1)

which satisfies the following equality:

cD R λ m , n F c [ f ( z ) ] +z [ D R λ m , n F c ( f ( z ) ) ] =(c+1)D R λ m , n f(z).
(3.2)

Theorem 3.1 Letc>1, 0ξ<1. Iff S λ m , n (ξ), then F c f S λ m , n (ξ).

Proof Let f S λ m , n (ξ). By using (3.2), we get

z [ D R λ m , n F c [ f ( z ) ] ] D R λ m , n F c [ f ( z ) ] =(c+1) D R λ m , n f ( z ) D R λ m , n F c [ f ( z ) ] c.

Let

z [ D R λ m , n F c [ f ( z ) ] ] D R λ m , n F c [ f ( z ) ] =ξ+(1ξ)h(z),h(z)=1+ c 1 z+ c 2 z 2 +.

We obtain

z [ D R λ m , n f ( z ) ] D R λ m , n f ( z ) ξ=(1ξ)h(z)+ ( 1 ξ ) z h ( z ) ξ + ( 1 ξ ) h ( z ) + c .

This implies

φ(μ,v)=(1ξ)μ+ ( 1 ξ ) v c + ξ + ( 1 ξ ) μ

(same as Theorem 2.1) and

Re { φ ( i μ 2 , v 1 ) } = ( 1 ξ ) ( c + ξ ) v 1 ( c + ξ ) 2 + ( 1 ξ ) 2 μ 2 2 , Re { φ ( i μ 2 , v 1 ) } ( 1 ξ ) ( c + ξ ) ( 1 + μ 2 ) 2 2 [ ( c + ξ ) 2 + ( 1 ξ ) 2 μ 2 2 ] < 0 .

After using Lemma 2.1 and Theorem 2.1, we have

F c f S λ m , n (ξ).

 □

Theorem 3.2 Let c>1 and ϕR with

Re { ϕ ( z ) } < c + ξ 1 ξ .

Iff S λ m , n (ξ,ϕ), then F c f S λ m , n (ξ,ϕ).

Proof Let f(z) S λ m , n (ξ,ϕ) and set

p(z)= 1 1 ξ ( z [ D R λ m , n F c [ f ( z ) ] ] D R λ m , n F c [ f ( z ) ] ξ ) ,
(3.3)

where p is analytic in U with p(0)=1.

Using (3.2) and (3.3), we have

(c+1) z [ D R λ m , n f ( z ) ] D R λ m , n F c [ f ( z ) ] =c+ξ+(1ξ)p(z).
(3.4)

Then, using (3.2), (3.3) and (3.4), we obtain

1 1 ξ ( z [ D R λ m , n f ( z ) ] D R λ m , n f ( z ) ξ ) =p(z)+ z p ( z ) ( 1 ξ ) p ( z ) + c + ξ .
(3.5)

Applying Lemma 2.2 to (3.5), we conclude that

F c f S λ m , n (ξ,ϕ).

 □

Author’s contributions

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

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The author thanks the referee for his/her valuable suggestions to improve thepresent article.

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Andrei, L. Some properties of certain subclasses of analytic functions involving adifferential operator. Adv Differ Equ 2014, 142 (2014). https://doi.org/10.1186/1687-1847-2014-142

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Keywords

  • analytic functions
  • differential operator
  • differential subordination
  • differential superordination