Two integral operators in the subclasses of -valent meromorphic functions

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Two integral operators in the subclasses of -valent meromorphic functions

Thamer Khalil M.S Al-Mustansirya University E-mail:[email protected]

Abstract: we defined several inclusion relationships in the punctured unit disc of two integral operators in the subclasses of -valent meromorphic functions, by the subclasses ( ) we derived some properties like, coefficient inequality, distortion bounds,growth and distortion bounds, convex set, Partial sums, radii of starlikeness and radii convexity.

Introduction. is the subclass of -valent meromorphic function of the form ( ) ∑

( * +) ( )

are analytic in * | | + is the punctured unit disc.

( ) : → is the integral operator [5] is deﬁned by:

( ) ( ) { ( ) ( )

∫ ( ) ( ) ⁽ ⁾ ( ) (2) And for ( ) we define the integral operator [9]

as follows:

( ) * ( ) ( ) ( ) ( )^{( )} ∫ ( ) ( ) ( ( ) ) ( ) for ( ) given by (1).

Now my combining two integral operators [8] ( ) and we obtain the linear operator

( ) as follows:

( ) ( ) ( ) . ( )/ ( ( ) ( )) ( )

Can be expressed the operator ( ) as follows:

( ) ( ) ( )

( ) ∑ ( ) ( )[

( )]

( )

τ ( ) ( ) ; ) By conditions (2) and (4), we define ( ) ( ) ( ) and

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( ) ( ) ( ) ( ) ( ) Definition (1): Given by (2) suppose .Then ( ) if can obtain the condition:

||

( ) . ( ) ( )/

( )

( ) ^.

( ) ( )/

( )

|| ( )

where z ; 0

Recently EL-Ashwah and another authors studied The meromorphic multivalent with different subclasses [6] [7] [1][2] [3][4] and [10].

Theorem (1): Let .Then is in the class ( ) iff

∑ ( )( )( )( )

( ) ( )

where z ; 0 In the function the result is sharp

( ) ( )

( )( )( )( ) ( ) Proof: Let (8) holds true, | | We give

|( ) . ( ) ( )/ ( ) |

|( ) . ( ) ( )/ ( )|

| ∑ ( ) ( )( )

|

| ∑ ( )

( )( ) ( )|

∑ ( )( )( )( )

( )

by hypothesis.

By maximum modulus principle, ( ) ( ) Conversely, let ( ) ( ) Then (7), we have

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|

( ) . ( ) ( )/

( )

. ( ) ( )/

( ) ( )

|

| ∑ ( )( )

∑ ( )( ) ( )| Since z then Real (z) | | obtain

{ ∑ ( )( )

∑ ( )( ) ( )} ( ) ( ( )) is real since the value of z chose on the real axis.

letting z , through real values (10) and by clearing the denominator of (10) we have

∑ ( )( )( )( )

( )

By setting the Sharpness of the result

( ) ( )

( )( )( )( ) ( ) ( ) Corollary (1): If ( ) ( ).Then

( )

( )( )( )( ) ( ) Theorem (2): suppose ( ) ( ) ( ) defined by

( ) ∑

( ) ( )

( ) ( ).Then the arithmetic mean of ( ) ( )

( ) ∑ ( ) ( )

is also in the class ( )

Proof: By (13), (14), write

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( )

∑( ∑

) ∑ (

∑

)

Since ( ). for every (i=1,2,…, ) using Theorem (1), we prove that

∑ ( )( )( )( )(

∑

)

∑(

∑ ( )( )( )( )

∑ ( ) ( )

Theorem (3): Define the partial sums (z) and (z) as follows (z) = and

( ) ∑

( * +) ( ) and suppose be given by (1) and let that

∑

(

( )( )( )( )

( ) ) ( )

So

( ( )

( )) ( ) ( ) and

( ( ) ( ))

( ) ( ) The possible (17) and (18) is the best for

Proof: We give by (16)

( ) then

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∑

∑ ∑

( )

Using

( ) ( ( )

( ) ( )) ∑

∑ ( )

consequently it is suffices

| ( )

( ) | ( ) ( )

And by (20), we have

| ( )

( ) | ∑

∑ ∑ ( )

if we take

( ) ( )

the bound in (17) is the best possible Similarly

( ) ( ) ( ( )

( ) ) ( ) ∑

∑ ( )

use (19), we obtain

| ( )

( ) | ( ) ∑

∑ ( ) ∑ ( ) by the assertion (18). The bounds given in the right of (18) is sharp for each with function (23).

Theorem (5): ( ) is convex set.

Proof: Let the arbitrary elements and of ( ) ( ).

Then for every ( ), we show that ( ) ( ) Thus, we have

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( ) ∑

,( ) -

Hence,

∑ ( )( )( )( )

,( ) -

( ) ∑ ( )( )( )( )

∑ ( )( )( )( )

( ) ( ) ( ) ( )

Theorem (6): ( ) is meromorphic -valent starlike of order ( ) in the disk

| | , if ( ) ( ) where

{ ( )

( )

( )( )( )( )

( ) } The result is sharp for the function ( ) by the condition (11).

Proof: It is sufficient

| ( )

( ) | | | ( ) Therefore

| ( ) ( )

( ) | |∑( )

∑ | ∑( ) | | ∑ | | Consequently (26), if

∑( ) | |

∑ | |

or

∑ ( )

| | ( )

Because ( ) ( ),then

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∑

( )( )( )( ) ( )

So, (27) true if

( )

| |

( )( )( )( ) ( )

or equivalent

| | { ( ) ( )

( )( )( )( )

( ) }

put 𝕽1=| |.

Theorem (7): (z) is meromorphic -valent convex of order ( ) in the disk

| | , if ( ) ( ) then

{ ( ) ( )( )( )( ) ( ) ( ) }

by (11) the result is sharp ( ).

Proof: to prove

| ( )

( ) | | | ( ) But

| ( )

( ) | | ( ) ( ) ( )

( ) | ∑ ( ) | | ∑( ) | |

Next, satisfied (28) if

∑ ( ) | |

∑( ) | |

or

∑ ( )

( ) | | ( ) Available online on http://www.rspublication.com/ijst/index.html Issue 9 volume 1 January- February 2019 ___________________________________________________________________________________________

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Since ( ) ( ), we have

∑

( )( )( )( )

( )

Then, (29) true if

( )

( ) | | ( )( )( )( ) ( ) or equivalent

| | { ( ) ( )( )( )( ) ( ) ( ) }

Putting 𝕽2=| |.

References

[1] W. G. Atshan, Subclass of meromorphic functions with positive coefficients defined by Ruscheweyh derivative II, J. Surveys in Mathematics and its Applications, 3 (2008), 67.

-77.

[2] W. G. Atshan and R.N Abdul-Hussien Some Properties of Certain subclass of Meromorphically Multivalent Functions Defined by Convolution and Integral Operator involving I-Function, College of Computer Science and Mathematics, University of Al- Qadisiya. Jul 30, 2014

[3] W. G. Atshan and S. R. Kulkarni, Meromorphic p-valent functionswith p

ositive coefficients defined by convolution and integral operator, IndianJournal of Acad.

Math. , 29(2) (2007), 409-423.

[4] W. G. Atshan and S. R, Kulkarni, Neighborhoods and partial sums of subclass of k- uniformly convex functions and related class of k-starlike functions with negative coefficients based on integral operators, Southeast Asian Bulletin of Mathematics, 33(4)(2009), 623-637.

[5] R.M. El-Ashwah, Properties of certain class of p-valent meromorphic functions associated with certain integral operator, Inf.Sci.Comput.,3(2014),pp.1-10.

[6] R.M. El-Ashwah, Certain class of meromorphic univalent func-tions deﬁned by an erdelyi-kober type integral operator, Open Sci.J. Math. Appl. 3 (1) (2015) 7–13.

[7] R.M. El-Ashwah, M.K. Aouf, Inclusion relationships of certainclasses of meromorphic p-valent functions, Southeast Asian Bull.Math. 36 (2012) 801–810.

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[8] R.M. El-Ashwah, A.H. Hassan, Properties of certain subclass of p-valent meromorphic functions associated with certain linear operator, Journal of the Egyptian Mathematical Society.,24(2016),pp.226-232.

[9] R.M. El-Ashwah, A.H. Hassan, Some inequalities of certain subclass of meromorphic functions deﬁned by using new integral operator, Inf. Sci. Comput. 3 (2014) 1–10.

[10]R.M.El-Ashwah, M.K.Aouf, A.A.H. Hassan, A.H.Hassan, Generaliztions of of hadamard product of certain meromorphic multivalent functions with positive cefficients,

Eur.J.Math.Sci.,2(1) (2013),pp. 41–50.