G0283341

IOSR Journal of Engineering (IOSRJEN) ISSN: 2250-3021 Volume 2, Issue 8 (August 2012), PP 33-41 www.iosrjen.org

On sandwich results for some subclasses of analytic functions involving certain linear operator Amnah Shammaky Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia Abstract : - The purpose of this paper is to derive some subordination and superordination results for certain normalized analytic functions in the open unit disk, acted upon by Carlson–Shaffer operator. Relevant connections of the results, which are presented in the paper, with various known results are also considered Keywords : - differential subordinations ; differential superordinations ; dominant ; subordinant I. INTRODUCTION Let H be the class of functions analytic in the open unit disk

  z : z  1 . Let H a, n be the

subclass of H consisting of functions of the form

f  z   a  a n z n  a n 1 z n 1  ...

Let A be the subclass of H consisting of functions of the form

f  z   a  a 2 z 2  ...

and we let





Am  f  H , f  z   z  am 1 z m 1  am  2 z m  2  ... .

With a view to recalling the principle of subordination between analytic functions, let the functions f and g be analytic in  .Then we say that the function f is subordinate to g if there exists a Schwarz function   z  , analytic in  with

 0  0 and z   1

z    , f z   g  z  z    .

such that We denote this subordination by

f  g or f z   g  z   z    .

In particular, if the function g is univalent in  , the above subordination is equivalent to f 0  g 0 and f    g   .

Let p, h  H and let  r , s, t ; z  : C    C . If p and if p satisfies the second-order superordination 3



  pz , zpz , z 2 pz ; z  are univalent and



hz    pz , zpz , z 2 pz ; z ,

(1)

f subordinate to F , then F is called to be superordinate to f .) An analytic function q is called a subordinant if q  p for all p satisfying (1). An ~ ~ for all subordinants q of (1) is said to be the best subordinant. univalent subordinant q that satisfies q  q Recently, Miller and Mocanu [7] obtained conditions on h , q and  or which the following implication holds hz     pz  , zp z , z 2 p z ; z   qz   pz  . then

p

is a solution of the differential superordination (1). (If

Using the results of Miller and Mocanu [7], Bulboaca[2] considered certain classes of firstorder differential superordinations as well as superordination-preserving integral operators [3]. Ali et al. [1] have used the results of Bulboaca [2] and obtained sufficient conditions for certain normalized analytic functions f  z  to satisfy zf z  q1 z    q2  z  , f z 

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On sandwich results for some subclasses of analytic functions involving certain linear operator

q1 and q 2 are given univalent functions in  with q1 0   1 and q 2 0  1. Shanmugam et al. [9] obtained sufficient conditions for a normalized analytic functions f  z  to satisfy where

q1 z  

f z   q2  z  , zf z 

and

where q1 and q 2 are given univalent functions in  with q1 0   1 and q 2 0  1.

While Obradovic and Owa [8] obtained subordination results with the quantity . A detailed investing- ation of starlike functions of complex order and convex functions of complex order using Briot – Bouquet differential subordination technique has been studied very recently by Srivastava and Lashin [10] (see also [11]) .

Let the function ϕ(a, c; z) be given by

a n z n 1 c 0,1,2,...; z    , n  0 c n 

 a, c; z   

where  x n is the Pochhammer symbol defined by

n  0;

xn  

1,

 xx  1x  2....x  n  1, n  N  1,2,3,....

Corresponding to the function  a, c; z  , Carlson and Shaffer [4] introduced a linear operator L(a, c), which is defined by the following Hadamard product (or convolution):

a n a z n 1 . n n  0 c n 

La, c  f  z    a, c; z   f  z    We note that

La, a  f z   f z , L2,1 f z   zf z  , L  1,1 f z   D  f z ,

where D f  z  is the Ruscheweyh derivative of 

f z  .

The main object of the present sequel to the aforementioned works is to apply a method based on the differential subordination in order to derive several subordination results involv- ing the Carlson–Shaffer Operator. Furthermore, we obtain the previous results of Srivastava and Lashin [10] and Obradovic´ and Owa [8] as special cases of some of the results presented here. q1 z 

z 2 f z   q z  ,  f z 2 2

II. PRELIMINARIES In order to prove our subordination and superordination results, we make use of the following known results.

Definition 2.1 [7 Definition 2, p. 817] Denote by Q the set of all functions f  z  that are analytic and injective on   E  f  , where

  E  f      : lim f z    , z    and are such that f    0 for     E  f  .

Theorem 2.2[6,Theorem 3.4h , p.132] Let the function q be univalent in the open unit disk  and  and

 be analytic in a domain D containing q  with     0 when   q   . Set Qz   zq z   qz  , hz    qz   Qz  . Suppose that (1) Q z  is starlike univalent in  , and www.iosrjen.org

34 | P a g e

On sandwich results for some subclasses of analytic functions involving certain linear operator (2) If

Re zhz  / Qz   0 for  z    .

 pz   zpz   pz    qz   zqz  qz  ,

then

pz   qz  and q is best dominant.

Lemma 2.3 [10] Let g be a convex function in  and let

hz   g z   m zg z  , where   0 and m is a positive integer. m If p  z   g 0   p m z  .... is analytic in  and pz   zp z   hz  , z    , then

p z   g  z  ,  z   

and this result is sharp.

Lemma 2.4 [9, Lemma 1, p,71] Let If p  H with

pz  

1



p0   a and

zp  z   h z ,

h be a convex function with h0  a and let   C with Re    0 .

 z   ,

then

p  z   q  z   h z   z   

where

qz  

 nz

 / n 

  ht t z

0

/ n 1

dt ,

z    .

The function q is convex and is the best dominant .

Theorem 2.5 [2] Let the function q be convex univalent in the open unit disk  and  and domain D containing q  . Suppose that

1 Re  qz  /  qz   0 for z    , 2 zqz  qz  is starlike univalent in  .

If p  H q0,1  Q , with p   D , and   pz   zp  z   p z  is univalent qz   zqz  qz   pz   zpz   pz  , then qz   pz  and q is the best subordinant .



be analytic in a

in  , and (2)

III. SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS We begin by proving involving differential subordination between analytic functions .

f z    , then zLa, c  f z   a La  1, c  f z  a  1La, c  f z  , c  0,1,2 ,...

Lemma3.1 If

where proof Note that

a n n 1 an z , n  0 c n 

La, c  f  z    and

a  1n n1 a z . c n n n 0 

La  1, c  f  z    These give that

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On sandwich results for some subclasses of analytic functions involving certain linear operator

a La  1, c  f  z  a  1La, c  f  z 

a n n1    a n n1    an a z   a 1 a z c n n c n n n 0 n 0  a    n  1 n a n z n 1 c n n 0   z La, c  f  z  

this proves lemma 3.1 .

q  z  be analytic and univalent in  such that qz   0 . zqz  / qz  is starlike univalent in  . Let     2 qz 2  zq z   zq z   0 z   ; ,  ,  ,   C;   0 Re1  qz    q z  qz    

Theorem 3.2 Let the function

Suppose that

(3) and      z  La, c  f  z            z    La  1, c  f  z      2  2   z    La, c  f  z         z La  1, c  f  z       a, c,  ,  ,  ,  , , f      La  1, c  f  z       a  La, c  f  z   1         La  2, c  f  z     1  a 1     La  1, c  f  z    

If

q

(4)

satisfies the following subordination :

zq z  qz  z   ; ,  ,  ,  ,  ,  C;   0;   0 ,

 a, c,  ,  ,  ,  , , f     qz   qz 2  

(5)

then

 z  La , c  f z      qz     z    La  1, c  f z   and q is the best dominant .

Proof Let the function p z  be defined by 



 La , c  f z   pz     z  



z   ; z  0 ;  ,  C;   0

  z    La  1, c  f z  

(6)



(7)

so that, by a straightforward computation , we have

 z La, c  f z    z La  1, c  f z   zp z  .    1   1  p z  La  1, c  f z    La, c  f z    

(8)

By using lemma3.1we deduce that

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On sandwich results for some subclasses of analytic functions involving certain linear operator

 a La  1, c  f z    1  a La  2, c  f z   . zp z     a    1  a    pz  La  1, c  f z     La, c  f z   By setting

         2 it can be easily observed

 

that    0 Also, by letting

  C \ 0 .

Qz   zq z  qz   

that

and

  

   

 , 

is analytic in C , 

 

zq z  qz 

is analytic in

and

(9)

and

hz    qz   Qz     qz    qz    2

we find that Q

C \ 0

zq z  , qz 

(10)

z  is starlike univalent in  and that

   zhz     2   Re1  qz   qz 2  zq z   zq z   0 Re  qz  q z    Q z     z   ; ,  ,  ,   C;   0 .

The assertion (6) of Theorem 3.2 now follows by an application of Theorem 2.2.

For

the

qz   1  Az / 1  Bz ,  1  B  A  1

choices

and

q z   1  z / 1  z , 0    1 , in Theorem 3.2, we get the following results (Corollaries 3.3 and 

3.4 ).

Corollary 3.3 Assume that (3) holds. If

f  A , and

1  Az   A  B z  1  Az   a, c,  ,  ,  ,  , , f         1  Az1  Bz 1  Bz  1  Bz  z   ; ,  ,  ,  ,  ,  C;   0;   0, 2

where

 a, c,  ,  ,  ,  , , f 

is

 z 1  Az  La , c  f z        z 1  Bz    La  1, c  f z   and 1  Az / 1  Bz is the best dominant.

Corollary3. 4 Assume that (3) holds. If f  A , and 



then 



as

defined

in

(4),

z  ; z  0;  ,  C;   0

1 z  1 z   a, c,  ,  ,  ,  , , f          1 z  1 z  z   ; ,  ,  ,  ,  ,  C;   0;   0

2



2  z 1 z2





a, c,  ,  ,  ,  , , f  is as defined in (4),     z  La , c  f z    1  z     then     z  ; z  0;  ,  C;   0 z 1  z     La  1, c  f z   where 

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On sandwich results for some subclasses of analytic functions involving certain linear operator and 1  z / 1  z is the best dominant. 

For a special case Corollary3. 5

q  z   e Az , with A   , Theorem 3.2 readily yields the following.

Assume that (3) holds. If

f  A , and

 a, c,  ,  ,  ,  , , f     e Az  e 2 Az  A z z   ; ,  ,  ,  ,  ,  C;   0;   0 where  a, c,  ,  ,  ,  , , f  is as defined in (4), then  z  La, c  f z      e Az z  ; z  0;  ,  C;   0    z    La  1, c  f z   Az and e is the best dominant .

Remark 3.6 Taking a  c  1,       0 ,   1,   1 /  in corollary 3.5, we get the 



result obtained by Obradovic and Owa [8].

For a special case

when q  z   1 / 1  z  b  C \ 0, a  c  1,       0 ,   1,     1and   1 / b , Theorem 3.2 reduces at once to the following known result obtained by Srivastava and Lashin [10].

Corollary3. 7 Let be a non-zero complex number. If f  A , and 2b

1  zf z   1  z 1   1  , b  f z   1 z

then

f z  1  z 1  z 2b

and 1 / 1  z 

2b

is the best dominant .

, a  c  1,       0 ,   1,   1 /  in Theorem 3.2, we get the For q  z   1  Bz  following known result obtained by Obradovic and Owa [8].

Corollary 3.8 Let  1  B  A  1. Let  , A, B satisfy the relation either   A B  / B

  A  B / B  1  1

then

or

 A  B / B  1  1. If f  A , and zf z  1  Az  , f z  1  Bz

 f z     A B  / B    1  Bz  z  

 z   ; z  0 ;   C;   0

and 1  Bz  is the best dominant.

With the help of Lemma 2.4, we now prove the following theorem .   A B  / B

Theorem 3.9 Let h  H , h0  1 , h0  0 which satisfy the inequality

 zhz  1 Re 1   ,  hz   2 

If

z .

f  Am satisfies the differential subordination www.iosrjen.org

38 | P a g e

On sandwich results for some subclasses of analytic functions involving certain linear operator

La  k , c  f z   h z  , z

k  Z





, z  ; z  0 ,

(11)

then

La  k  1, c  f z   g z  , z where

g z   The function

g

a  k  1 mz

a  k 1 / m 

  ht t z

a  k 1 / m 1

0

k  Z dt ,





, z  ; z  0 ,

k  Z



(12)



, z .

(13)

is convex and is the best dominant .

pz  be defined by La  k  1, c  f z  p z   k  Z  , z  ; z  0 . z

Proof Let the function





(14)

A straightforward computation gives

  zpz   z La  k  1, c  f z     1 . pz   La  k  1, c  f z   

(15)

By using lemma3.1we deduce that

 zp z   a  k  1La  k , c  f z    a  k  1 pz   La  k  1, c  f z  

and hence

pz  

zp z  La  k , c  f z   a  k 1 z

k  Z





, z .

(16)

The assertion of Theorem 3.9 now follows from Lemma 2.4.

For the choice of k=1, we get

Theorem 3.10 If

f  Am satisfies the differential subordination

La  1, c  f z   hz  , z then

La , c  f z   g z  , z

where

g z   The function Proof

 z  ; z  0 ,

g

1  a  mz

a / m 

  ht t z

0

a / m 1

(17)

 z  ; z  0 , dt ,

 z   .

(18)

(19)

is convex and is the best dominant .

pz  be defined by La , c  f z  p z   z

Let the function

 z  ; z  0 ,

(20)

so that, by a straightforward computation , we have

  zp z   z La, c  f z     1 . pz   La, c  f z   

(21)

By using lemma3.1we deduce that

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39 | P a g e

On sandwich results for some subclasses of analytic functions involving certain linear operator

 zp z   a La  1, c  f z    a pz   La , c  f z   and hence

p z  

zpz  La  1, c  f z   a z

 z.

(22)

The assertion of Theorem 3.10 now follows from Lemma 2.4.

Next, by using Lemma 2.3, we prove the following theorem .

Theorem 3.11 Let

g

g 0  1.

be a convex function with

hz   g z  

Let

h be a function , such that

m zg z  .  1

f  Am satisfies the differential subordination La  k , c  f z   hz , k  Z  , z  ; z  0 , z

If





(23)

then

La  k  1, c  f z   g z  , k  Z  , z  ; z  0 , z

(24)

and this result is sharp.

Proof The proof of the theorem is much akin to the proof of Theorem 3.12 and hence we omit the details involved.

Next, by appealing to Theorem 2.5 of the preceding section, we prove Theorem 3.12.

q  z  be analytic and convex univalent in  such that qz   0 and zqz  / qz  starlike univalent in  . Further, let us assume that  2   2 Re qz   qz   0 , z   ;  ,  ,   C;   0 (25)   

Theorem 3.12 1 Let be



If



f  A , 0   La , c  f z    

 z   H q0 ,1  Q ,   La  1, c  f z  

z



and

 a, c,  ,  ,  ,  , , f  is univalent in  , then   q z    q z 2  

zq  z  q z 

  a, c,  ,  ,  ,  , , f 

z   ; ,  ,  ,  ,  ,  C;   0;   0, implies

 La, c  f z   qz     z  





  z    H q0,1  Q  La  1, c  f z  

and q is the best subordinant where  Proof By setting

a, c,  ,  ,  ,  , , f  is as defined

         2 it is easily observed that

(26)

and

   

 , 

   is analytic in C . Also,        0 ,   C \ 0 .

is analytic in C

www.iosrjen.org

in (4).

\ 0 and that 40 | P a g e

On sandwich results for some subclasses of analytic functions involving certain linear operator Since

q

is convex (univalent ) function it follows that ,

 qz   2   2 Re  Re qz   qz   0 , z   ;  ,  ,   C;   0 .       qz   The assertion (26) of Theorem 3.12 follows by an application of Theorem 2.5.

We remark here that Theorem 3.12 can easily be restated , for different choice of the function q Theorem 3.2 and Theorem 3.12, we get the following sandwich theorem .

q1 z  be convex univalent q 2 z   0 . Suppose f satisfies (25)and (3) .

Theorem 3.13 Let

and that then



in



such that

q1 z   0 and

 z   H q0,1  Q ,   La  1, c  f z  

z



a, c,  ,  ,  ,  , , f  is univalent in  , where  a, c,  ,  ,  ,  , , f  is given by (4)

  q1 z    q1 z 2   zq  z   2 q 2 z 

z  be univalent



f  A , 0   La , c  f z    

If

and q 2

z  . Combining

zq1 z  2   a, c,  ,  ,  ,  , , f      q 2 z    q 2 z   q1 z 

z   ; ,  ,  ,  ,  ,  C;   0;   0,

 La , c  f z   q1 z     z  





  z    q 2 z   La  1, c  f z  

implies

z   ; z  0 ;  ,  C ;   0 and q1

and

q 2 are respectively the best subordinant and best dominant.

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