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Elmergib University

Faculty of Art & Science Kasr Khiar

Journal of Humanitarian, Scientific

and Social Sciences

3rd

Issue, June 2017

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Journal of Humanitarian, Scientific and Social Sciences

3rd

Issue June 2017

DIFFERENTIAL SANDWICH THEOREMS FOR P-VALENT

FUNCTIONS DEFINED BY CONVOLUTION INVOLVING

CERTAIN FRACTIONAL DERIVATIVE OPERATOR

Somia M. Amsheri

Department of Mathematics, Faculty of Science,

Elmergib University, AL Khums, Libya

[email protected]

Abstract

In the present paper we derive some subordination and superordination results

for p-valent functions in the open unit disk defined by convolution involving

certain fractional derivative operator. Relevant connections of the results, which

are presented in the paper, with various known results are also considered.

Keywords: p-valent function, Hadamard product, incomplete beta function,

fractional derivative operators, Differential subordination, Differential

superordination,.

Mathematics subject classification: 30C45, 26A33

1. Introduction and Preliminaries

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Let denote the class of analytic functions in the open unit disk

| | and let denote the subclass of the functions of the

form:

Also, let be the class of functions of the form

∑

and set . For functions , given by (1.1), and given by

∑

The Hadamard product (or convolution) of and is defined by

∑

Let , we say that the function is subordinate to , if there exist a

Schwarz function , analytic in , with and | | ,

such that for all .

This subordination is denoted by or . It is well known that, if

the function is univalent in , then if and only if

and .

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Let , and let . If and

are univalent functions, and if satisfies the

second-order superordination

then is called to be a solution of the differential superordination . (If

is subordinatnate to then is called to be superordinate to ).

An analytic function is called a subordinant if for all

satisfies . An univalent subordinant ̃ that satisfies ̃ for all

subordinants of is said to be the best subordinant.

Recently, Miller and Mocanu obtained conditions on and for

which the following implication holds true:

with the results of Miller and Mocanu , Bulboaca investigated certain

classes of first

order differential superordinations as well as superordination-preserving integral

operators

. Ali et al. used the results obtained by Bulboaca and gave the sufficient

conditions for certain normalized analytic functions to satisfy

where and are given univalent functions in with and

. Shanmugam et al. obtained sufficient conditions for a normalized

analytic functions

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to satisfy

and

where and are given univalent functions in with and

.

Let be the Gauss hypergeometric function defined for by,

(see Srivastava and Karlsson )

∑

where is the Pochhammer symbol defined, in terms of the Gamma function,

by

{

for

We recall the following definitions of fractional derivative operators which were

used

by Owa , (see also ) as follows:

Definition 1.1 The fractional derivative operator of order is defined by,

∫

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where is analytic function in a simply- connected region of the

z-plane containing the origin, and the multiplicity of is removed by

requiring log ( to be real when

Definition 1.2 Let , and . Then, in terms of the familiar

Gauss’s hypergeometric function F , the generalized fractional derivative

operator

is

(

∫ (

) )

where is analytic function in a simply- connected region of the z-plane

containing the origin with the order | | , where

and the multiplicity of is removed by requiring

to be real when

Definition 1.3 Under the hypotheses of Definition 1.2, the fractional derivative

operator

of a function is defined by

Notice that

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With the aid of the above definitions, we define a modification of the fractional

derivative operator

by

for and . Then it

is observed that

maps onto itself as follows:

∑

Let be the incomplete beta function defined for by

∑

where and is given by (1.6), and using the

Hadamard product, we define the following operator

by

If , then from and , we can easily see that

∑

where

and .

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Notice that, if , then we have

and

. Also, we observe that the operator

reduces

to the following operators considered earlier for different choices of and

.

(1) For , we get the operator which is motivated from

Carlson-Shaffer operator [5].

(2) For and , we get the operator

which is the Ruscheweyh derivative operator of order (see [12,

15]).

It is easily verified from (1.15) that

(

)

The object of this paper is to derive several subordination and superordination

results defined by Hadamard product involving certain fractional derivative

operator. Furthermore, we obtain the previous results of Aouf et al. [1],

Shammugam et al. [14], Obradovic et al. [8], Obradovic and Owa [9], and

Srivastava and Lashin [16] as special cases of some the results presented here.

In order to prove our results we mention to the following known results which

shall be used in the sequel.

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Lemma 1.4 [11] Let , such that and .

Then

Definition 1.5 [7] Denoted by the set of all functions that are analytic and

injective in ̅ where

{

}

and are such that for .

Lemma 1.6 [6] Let the function be univalent in the open unit disk , and

and be analytic in a domain containing with when

. Set and Suppose that

1. is starlike univalent in , and

2. (

) for

If

( ) ( )

Then and is the best dominant.

Lemma 1.7 [3] Let the function be univalent in the open unit disk , and

and be analytic in a domain containing with when

. Suppose that

1. (

) for

2. ( ) is starlike univalent in .

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If with , and ( ) is

univalent in , and

( ) ( )

then and is the best subordinant.

2. Subordination and superordination for p-valent functions

We begin with the following result involving differential subordination between

analytic functions.

Theorem 2.1. Let (

)

and let the function be analytic

and univalent in such that . Suppose that

is starlike

and univalent in . Let

{

( )

}

and , and

(

)

(

)

[

]

If satisfies the following subordination:

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( )

then

(

)

and is the best dominant.

Proof. Let the function be defined by

(

)

So that, by a straightforward computation, we have

[

(

)

]

By using the identity we obtain

[

]

By setting and

, it can be easily verified that

is analytic in , is analytic in , and that .

Also, by letting

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( )

and

( ) ( )

we find that is starlike univalent in and that

{

} {

( )

}

The assertion of Theorem 2.1 now follows by an application of Lemma 1.6.

Remark 1. For the choices

and

(

)

, in Theorem 2.1, we get the following results (Corollary 2.2

and Corollary 2.3) below.

Corollary 2.2. Assume that (2.1) holds. If , and

(

) (

)

where is as defined in (2.2), then

(

)

and

is the best dominant.

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Corollary 2.3. Assume that (2.1) holds. If , and

(

)

(

)

where is as defined in (2.2), then

(

)

(

)

and (

)


Remark 2. Taking and in Theorem 2.1, Corollary

2.2 and Corollary 2.3, we obtain the subordination results for linear operator due

to Shammngam et al. [[14], Theorem 3, Corollary 1 and Corollary 2,

respectively].

Remark 3. For the choice

and

in Theorem 2.1, we obtain the

following known result due to Obradovic et al. [8].

Corollary 2.4. Let such that | | or | | If

, and

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(

)

then

(

)

and


Remark 4. For , Corollary 2.4 reduces to the recent result of Srivastava

and Lashin [16].


and in Theorem 2.1, we obtain the following known result

due to Obradovic and Owa [9].

Corollary 2.5. Let with and suppose that |

|

or |

| . If , and

(

)

then

(

)

and


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| |

and

in Theorem 2.1, we

get the following result due to Aouf et al. [1].

Corollary 2.5. Let and | |

, and suppose that

| | or | | . Let , and suppose

that

for all . If

(

)

then

(

)

and


Next, by appealing to Lemma 1.7 of the preceding section, we prove the

following.

Theorem 2.7. Let be analytic and univalent in such that and

be starlike and univalent in . Further, let us assume that

{

( )

}

If ,

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(

)

and is univalent in , then

( )

implies

(

)

and is the best subordinant where is as defined in .

Proof. By setting and

, it can be easily

observed that is analytic in , is analytic in , and that

. Since is convex (univalent) function it follows that,

{

} {

( )

}

The assertion of Theorem 2.7 now follows by an application of Lemma 1.7.

Combining Theorem 2.1 and Theorem 2.7, we get the following sandwich

theorem.

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Theorem 2.8. Let and be univalent in such that and

with

and

being starlike univalent. Suppose that satisfies

(2.4) and satisfies (2.1). If ,

(

)

and is univalent in , then

( )

( )

implies

(

)

and and are respectively the best subordinant and the best dominant where

is as defined in .

Remark 7. Taking and in Theorem 2.6 and

Theorem 2.7, we obtain the superordination and sandwich results for linear

operator due to Shammngam et al. [[14], Theorem 4 and Theorem 5,

respectively].

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Remark 8. For and in Theorem 2.8, we get the following

result.

Theorem 2.9. Let and be univalent in such that and

with

and

being starlike univalent. Suppose that

satisfies (2.4) and satisfies (2.1). If ,

(

)

and let

(

)

(

)

(

)

is univalent in , then

( )

( )

implies

(

)

and and are respectively the best subordinant and the best dominant.

References

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[1] Aouf, M. K., AL-Oboudi, F. M., and Haidan, M. M., 2005, “On Some Results

for -Apiralike and -Robertson Functions of Complex Order”, Publ. Inst.

Math. (Belgrade), 77(91), pp. 93-98.

[2] Ali, R. M., Ravichandran, V., Hussain, K. M., and Subramaniant, K. G., 2005,

“Differential Sandwich Theorems for Certain Analytic Functions”, Far East J.

Math. Sci. 15(1), pp. 87-94.

[3] Bulboaca, T. A., 2002, “Classes of First-Order Differential Superordinations”,

Demonstr. Math. 35(2), pp. 287-292.

[4] Bulboaca, T. A., 2002, “A Class of Superordination-Preserving Integral

Operators”, Indag. Math. New Ser. 13(3), pp. 301-311.

[5] Carlson, B. C., and Shaffer, D. B., 1984, “Starlike and Prestarlike

Hypergeometric Functions”, SIAM, J. Math. Anal, 15(4), pp. 737-745.

[6] Miller, S. S., and Bulboaca, T. A., 2000, “Differential Subordinations: Theory

and Applications”, Pure and Applied Mathematics, No. 225, Marcel Dekker,

New York.

[7] Miller, S. S., and Mocanu, B. T., 2003, “Subordinations of Differential

Superordinations”, Complex Variables, 48(10), pp. 815-826.

[8] Obradovic, M., Aouf, M. K., and Owa, S., 1989, “On Some Results for

Starlike Functions of Complex Order”, Publ. Inst. Math. (Beograd) (N. S.),

46(60), PP. 79-85.

[9] Obradovic, M., and Owa, S., 1990, “On Certain Properties for Some Classes

of Starlike Functions”, J. Math. Anal. Appl., 145, pp. 357-364.

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[10] Owa, S., 1978, “On The Distortion Theorems- I”, Kyungpook. Math. J. 18,

pp. 53-59.

[11] Raina, R. K., and Srivastava, H. M., 1996, “A Certain Subclass of

Analytic Functions Associated With Operators of Fractional Calculus”,

Computers & Mathematics with Applications, 32, pp. 13-19.

[12] Ruscheweyh, S., 1975, “New Crieteria for Univalent Functions”, Proc.

Amer. Math. Soc. 49, pp. 109-115.

[13] Shanmugam, T. N., Ravichandran, V., and Sivasubramanian, S., 2006,

“Differential Sandwich Theorems for Some Subclasses of Analytic Functions”,

Austral J. Math. Anal. Appl. 3(1), pp. 1-11.

[14] Shanmugam, T. N., Ramachandran, G., Darus, M., and Sivasubramanian,

S., 2007, “Differential Sandwich Theorems for Some Subclasses of Analytic

Functions Involving a Linear Operator”, Acta. Math. Univ. Comenianae. Vol.

LXXVI, (2), pp. 287-294.

[15] Srivastava, H. M., and Owa, S., Editors, 1992, “Current Topics in Analytic

Function Theory”, World Scientific, Singapore.

[16] Srivastava, H. M., and Lashin, A. Y., 2005, “Some Applications of the

Briot-Bouquet Differential Subordination”, J. Inequal. Pure Appl. Math. 6(2),

Article 41, 7pp.

[17] Srivastava, H. M., and Karlsson, P. M., 1985, “Multiple Gaussian

Hypergeometric Series”, Halsted Press (Ellis Horwood Limited, Chichester),

Wiley, New York/ Chichester/ Brishane/ Toronto.

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