



































THE  D  - OPERATOR AND GAUSS FUNCTIONS OF THREE VARIABLS


  

Effectiveness similar transposed set of polynomials  

of two complex variables in different regions 
Mosaed M. Makky and Mohamed O. Soltan  

Department of Mathematics, Faculty of Science, South Valley University (Qena-Egypt) 
Email: mosaed_makky11@yahoo.com, mosaed_makky@sci.svu.edu.eg 

mohamed.abuelhassan2015@gmail.com 
Abstract  
In this paper we derive the effectiveness of similar transposed sets of polynomials of 

two complex variables in origin, when the constituent sets are originally effective under 

a normalizing conditions for these sets. 

Moreover, when the constituent sets under the normalizing conditions are algebraic and 

functional sets, the effectiveness of similar transposed sets of polynomials in open 

hyperspheres is given here. Finally the effectiveness of similar transposed sets of 

polynomials and effectiveness of inverse similar transposed sets of polynomials are 

studied here. 

   

2010 Mathematics Subject Classification: Primary 30C10, 30A10, Secondary 30B10. 

Keywords: Similar sets, transposed sets, inverse transposed sets, basic sets, Canon sum, 

Cannon function. 

1. Introduction and Preliminaries 
In recent decades, we have centered our attention on a new family of multivariate 

polynomials, and similar sets of polynomials, which are a great example of using 

operational techniques in a general setting. 

So we will first present similar sets of polynomials and then summarise some basic 

findings related to these two families of bivariate polynomials which provide the main 

background in our analysis along with the principle of analytical functions in [4, 10]. 

Polynomial sequences play an important role in solving numerous problems that exist in 

many different fields of pure and applied mathematics (see, for example, [3, 11, 22]).  

In 1937 Cannon [2] introduced convergence of some polynomials among the many 

polynomials. 

The main objective of this paper is to study the effectiveness similar transposed sets of 

polynomials of one complex variable, which was recently defined and studied by 

Sayyed and Mena [18, 19], Sayyed and Metwally [20, 21]. 

Newns [15] was introduced the transposed inverse set of a given basic set of 

polynomials is the set whose matrix of coefficients is the transposed inverse of that of 

the given set. Adepoju [1; Chapter II]), introduced the effectiveness properties, in Faber 

regions, of the transposed inverse set of a given basic set of polynomials. 

In addition, similar sets of polynomials of two complex variables were defined and 

studied by Makky [8, 9], a sequence  , ( , )m np z w  being a basic set of polynomials of 

monomialvariables z and w is said to form a basic set, if thecomplextwo

; , 0m nz w m n   for a unique finite representation as follows (see [2,5, 13]): 

(1.1)     
( , )

, ; , ,

( , ) 0

( , )
m n

m n

m n h k h k

h k

z w p z w


    

and the polynomials  , ( , )m np z w are expressed in polynomial form as follows: 

(1.2)   
( , )

, , ; ,

( , ) 0

( , )
m n

h k

m n m n h k

h k

p z w p z w


  . 

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mailto:mosaed_makky11@yahoo.com
mailto:mosaed_makky@sci.svu.edu.eg


  

The values ,

,

h k

m np  and ,

,

h k

m np  are called matrices of coefficients and operators of the basic 

set  , ( , )m np z w  respectively; each of which is row finite. Thus, the necessary and 

sufficient condition for the set  , ( , )m np z w to be basic if 

 , ,

, ,

m n m n

m n m np p I   

where I is an infinite unit matrix and  (m,n) =
1

2
(m+n)(m+n+1)+n. 

Let  ( )

, ( , ) ; 1,2i

m np z w i  where 
( , )

( ) ,

, ,

( , ) 0

( , ) ; 1,2
m n

i h k h k

m n m n

h k

p z w p z w i


   basictwo

sets of polynomials of two complex variables be. Also, the matrices coefficients and 

operators    ( ) ( ) ,

,

i i h k

m np p ,  ( ) ( ) ,

,

i i h k

m np p  are arranged according to the sequence of 

double suffices entities 
,( )i je followsas

0,0 1,0 0,1 2,0 1,1 0,2, , , , , , .....e e e e e e  ; valuethe

( , )i j for the enumerator number of 
,i j among this sequence, such that: 

  
1

( , ) ( )( 1) ; ( , ) 0
2

i j i j i j j i j      . 

The basic set  , ( , )m np z w of polynomials will be called simple set if the polynomials 

, ( , )m np z w  are of order n, if 

  
( , )

,

, ,

( , ) 0

( , )
m n

h k h k

m n m n

h k

p z w p z w


   

and it is a monic set if ,

, 1m n

m np    for all (m,n), a basic set  , ( , )m np z w of polynomials is 

said to be Cannon set, if the number; 
,m nN ; of non -zero elements in the relation (1.1) 

holds   
1

,lim { } 1m n
m n

m n
N 

 
  ,  

otherwise it is called a general basic set (see e.g. [2]). 

Also, the basic set  , ( , )m np z w is said to be algebraic of degree N; when its matrix of 

coefficients satisfies the usual identity in [13] as follows:  

  1

0 1 ... 0N N

Na p a p a I    . 

The Cannon sum 
, [ ]m n r ; of the general basic set , ( , )m np z w is given by (see [14, 16, 

17]) 

(1.3)  
, [ ]m n r = 

( , )
,

, , ,

( , ) 0

| | ;
m n

h k

m n m n m n

h k

p M p r


    

and the Cannon function for the same set is   

(1.4)   
1

,[ ] limsup { ( , )}m n
m n

m n

r z w  

 

 . 

thatsupposeAlso,  , ( , )m np z w settheofpolynomialsofsetinversebe

 , ( , )m np z w where  

(1.5)  
( , )

,

, ,

( , ) 0

( , )
m n

h k h k

m n m n

h k

p z w p z w


   

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and  
( , )

,

, ,

( , ) 0

( , )
m n

m n m n

h k h k

h k

z w p p z w


  . 

Let  ( )

, ( , ) ; 1,2i

m np z w i   are two basic sets of polynomials and set  , ( , )m np z w  is 

called the product set of two sets  ( )

, ( , ) ; 1,2i

m np z w i   (see [1, 11, 12, 13]),  

     (1) (2)

, , ,( , ) ( , ) ( , )m n m n m np z w p z w p z w , 

( , ) ( , ) ( , )
, (1) , (2) ,

, , , ,

( , ) 0 ( , ) 0 ( , ) 0

( , )
m n m n h k

m n h k s t h k h k

m n h k m n s t

h k h k s t

p z w p z w p p z w
  

    . 

Makky in [9] study effectiveness the similar sets of polynomials of a single complex 

variable when each of the constituent sets is basic.  

Now, consider similar sets of polynomials of two complex variables, whenever each of 

the constituent sets are transposed basic sets. 

Definition: 

Assume that  ( )

, ( , ) ; 1,2i

m np z w i   be a transposed basic sets of polynomials; and let 

 , ( , )m nu z w a basic set of polynomials given by (see [14, 18]) 

(1.7)       (1) (2) (1)

, , , ,( , ) ( , ) ( , ) ( , )m n m n m n m nu z w p z w q z w p z w  

where 

(1.8)   
( , )

, , ; ,

( , ) 0

( , )
m n

h k

m n m n h k

h k

u z w u z w


    

and  
( , ) ( , ) ( , )

, (1) , (2) , (1) ,

, , , ,

( , ) 0 ( , ) 0 ( , ) 0

m n s t i j
h k s t i j h k h k

m n m n s t i j

s t i j h k

u p p p z w
  

    . 

That can also, be written in the form  
( , ) ( , ) ( , )

, (1) , (2) , (1) ,

, , , ,

( , ) 0 ( , ) 0 ( , ) 0

m n s t i j
h k m n s t i j h k

m n s t i j h k

s t i j h k

u p p p z w
  

    . 

Then the set  , ( , )m nu z w  is called a similar transposed set of polynomials of two 

complex variables (see .e.g. [7]).  

similar transposed sets while theybasic property forConfiguring the

 ( )

, ( , ) ; 1,2i

m np z w i   are basic. Also, let ( ) ; 1,2ip i  , are matrices of coefficients 

of the sets  ( )

, ( , ) ; 1,2i

m np z w i  ,  ( ) ( ) ,

,

i i h k

m np p  and the values U , U  are matrices 

coefficients of the similar transposed sets  , ( , )m nu z w .  

Write the matrices (1) (2) (1)U p p p  and (1) (2) (1)U p p p , then we get  
(1) (2) (1) (1) (2) (1)UU p p p p p p I    

and 
(1) (2) (1) (1) (2) (1)U U p p p p p p I   

where I is unit infinite matrix. 

Hence the matrix U of coefficients of the set   , ( , )m nu z w  has a unique inverse U , 

therefore the set  , ( , )m nu z w is basic. 

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2- Effectiveness of similar transposed set of polynomials at the origin   
In this section we study the effectiveness of a similar transposed set of polynomials 

 , ( , )m nu z w  two complex variables, at the origin with normalizing conditions and 

 ( )

, ( , ) ; 1,2i

m np z w i   fulfil the following conditions: 

 (2.1)  ( )[0 ] 0i    

(2.2)  ( )[0 ] 0; 0i r     

where  

(2.3)   
1

( ) ( )

, ,[0 ] limsup ,i i m n
m n m n

m n

M p r  

 

     , 

(2.4)   
1

( ) ( )

, ,[0 ] liminf ,i i m n
m n m n

m n
M p r  

 
      

whenever  

  ( ) ( )

, ,, max | ( , ) |
r

i i

m n m n
S

M p r p z w    . 

Therefore the transposed sets  ( )

, ( , ) ; 1,2i

m np z w i   satisfy the conditions: 

(2.5)  ( )[0 ] 0i    

(2.6)  ( )[0 ] 0; 0i r    

where  

(2.7)  

1

( ) ( )

, ,

1
[0 ] limsup ,

m n
i i

m n m n
m n

M p
r

 


 

  
    

  
, 

(2.8)  

1

( ) ( )

, ,

1
[0 ] liminf ,

m n
i i

m n m n
m n

M p
r

 


 

  
    

  
 

and  

  ( ) ( )

, ,

1
, max | ( , ) |

r

i i

m n m n
S

M p p z w
r

 
 

 
. 

Also, the transposed inverse set  (1)

, ( , )m np z w  satisfy the following conditions: 

(2.9)  (1)[0 ] 0    

(2.10)  

1

(1) (1)

, ,

1
[0 ] limsup ,

m n

m n m n
m n

M p
r

 


 

  
    

  
, 

To study the effectiveness of similar transposed set of polynomials at the origin, we 

present at the beginning some lemmas that explain this experiment in preparation to 

prove this effectiveness. 

Lemma (2.1):  

Following set  ( , )jp z w  satisfies the condition (2.1), then the transposed power set 

 ( , )m

jp z w  satisfies the condition [0 ] 0m   . 

Proof: 

By transposed product set,  write the square transposed set  2 ( , )jp z w  where   

     2 ( , ) ( , ) ( , )j j jp z w p z w p z w  

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then the square set is the transposed product set of two coincident sets, each of which 

satisfies the condition (2.5), then the set  2 ( , )jp z w  satisfies condition 2 [0 ] 0   . 

power settheTherefore  ( , )m

jp z w  satisfies condition [0 ] 0m    setand the

 1 ( , )m

jp z w  is transposed product of two sets  ( , )jp z w  and  ( , )m

jp z w  as 

follows: 

     1 ( , ) ( , ) ( , )m m

j j jp z w p z w p z w   

which satisfies respective conditions (2.5) and [0 ] 0m   , then the power set 

 1 ( , )m

jp z w  conditionsatisfies  1 [0 ] 0m     and lemmathisthatofproofthe

follows, by induction.  

Lemma (2.2):  

Following transposed sets  ( )

, ( , ) ; 1,2i

m np z w i   satisfy the condition (2.5) and the set 

 (1)

, ( , )m np z w  is algebraic one, then the similar transposed set  , ( , )m nu z w  holds: 

(2.11)    
1

, ,[0 ] limsup max ( , ) 0
r

m n

m n mn
Sm n

u z w 


 

   . 

Proof: 

Let the set  (1)

, ( , )mnp z w  satisfies the condition (2.5) and by lemma (2.1), it follows that 

power set  (1)

, ( , )j

m np z w  accords the condition 
(1)

, [0 ] 0j

m n   . Hence by (2.7) we get  

(2.12)  (1)

, , 1 1 2, ; , 0, 1.j m n

m n m nM p r k r m n j       

If the set  (1)

, ( , )m np z w  is an algebraic set then we get  

(2.13)  1( ) , , ( ) ,

, , , 0 ,1

it ti h k h k j i h k

m n m n m n j m nj
p p   


   

where  
0 ; 1, 2it i   are constituent and  (1)

, ( , )j

m np z w ;    1j   is the j-th power of 

transposed set  (1)

, ( , )m np z w .  

By (2.12) and using Cauchy's inequality, the relation (2.13) is  

(2.14)  
,( ) , 6

, 1 1 1

, 5

( 1)
m n

h ki h k

m n h k

m n

r
p k t

r









   

where   
0 1
max | |j

j


 
. 

Suppose that  (2)

, ( , )m np z w satisfies (2.5), we have  

(2.15)  
(2)

, , 1

5 4

1 1
, ; , 0m n m n m n

M p k m n
r r




 
 

 
. 

By relations (2.14), (2.15) and Cauchy's inequality we get  

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1 7

, ,

7

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , ,

( , ) 0 ( , ) 0 ( , ) 0 7 ,

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , ,

( , ) 0 ( , ) 0 ( , ) 0

1
(2.16) [ ; ] max ( , )

1
| || || |

| || ||

r

m n m n
S

m n h k i j
h k i j s t

m n h k i j s t
h k i j s t s t

m n h k i j
h k i j i j

m n h k s t

h k i j s t

M u u z w
r

p p p
r

p p p


  

  







  

  
7 ,

1
|

s t

s tr 

 

( , ) ( , ) ( , )
,(1) , (2) , 6

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 5 7 ,

( , ) ( , ) ( , )
, 6 ,(1) , (2) ,

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 5 , 7

1
| || |

1
| || |

s tm n h k i j
i jh k i j

m n h k i j s t
h k i j s t s t s t

s tm n h k i j
i j i jh k i j

m n h k i j s t
h k i j s t i j s t

r
k k p p

r r

r
k k p p

r r



 

 

 



 
  



 
  

  

  
,s t

 

( , ) ( , ) ( , )
, 6 ,(1) , (2)

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 5 , 7 ,

( , ) ( , ) ( , )
, 6 ,(1) ,

1 2 ,

( , ) 0 ( , ) 0 ( , ) 0 4 , , 7 ,

1
| | [ , ]

1
| |

s tm n h k i j
i j i jh k

m n h k s t
h k i j s t s t s t

s tm n h k i j
i j i jh k

m n h k s t
h k i j s t h k s t s t

r
k k p M p

r r

r
k k p

r r

 

 

 

  




  



 
  

  

  

 

( , ) ( , ) ( , )
, 6 ,(1) (1) ,

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 04 4 , , 7 ,

(1)

1 2 ,

4

1 1
[ , ] | |

1
[ , ]

s tm n h k i j
i j i jh k

h k m n h k s t
h k i j s t h k s t s t

m n

r
k k M p p

r r r

k k M p
r

 

  



 
  

  
 

from which, we obtain  
11

(1) (1)

, 1 2 ,

7 7 4 4

1 1 1 1
sup [ ; ] sup ,

m nm n

m n m n
m n m n

Lim M u Lim k k M P
r r r r

 


   

         
          

         
 

where    2 1 1( 1)k t  . 

If  4 7r and r  are chosen near to r then we get  

   (1)0 0 0     . 

Therefore we obtain   

   0 0   , 

Now, we can use the lemmas above to prove the following theorem concerning the 

effectiveness of  similar transposed set  , ( , )m nu z w  of two complex variables at the 

origin: 

Theorem (2.1): 

Let  ( )

, ( , ) ; 1,2i

m np z w i   be two algebraic sets satisfy condition (2.5), and then the 

similar transposed set  , ( , )m nu z w will be effective at the origin. 

Proof: 

Since each of two sets  ( )

, ( , ) ; 1,2i

m np z w i   is algebraic condition (2.5) satisfied  

(2.17)  
,(1) , 4

, 1 1 1

, 3

( 1)
m n

h kh k

m n h k

m n

r
p k t

r









  

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(2.18)  
,(2) , 2

, 1 2 2

, 1

( 1)
m n

h kh k

m n h k

m n

r
p k t

r









 . 

Also, the sets  ( )

, ( , ) ; 1,2i

m np z w i   satisfy the condition (2.11), then we get 

   0 0   . 

Therefore  

(2.19)   
, , 1

6 5

1 1
[ ; ]m n m n m n

M u k
r r




. 

Inserting (2.16), (2.17). (2.18) and using Cauchy's inequality in Cannon sum of similar 

transposed set  , ( , )m nu z w we get: 

( , )
,

, , , ,

( , ) 06 6

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , , ,

( , ) 0 ( , ) 0 ( , ) 0 5 ,

1 1
[ ] [ ; ]

1
| || || |

m n
s t

m n m n m n s t

s t

m n h k i j
h k i j s t

m n m n h k i j s t
h k i j s t s t

u M u
r r

p p p
r









  

 





  
 

( , ) ( , ) ( , )
(1) , (2) , (1) ,

1 , , , ,

( , ) 0 ( , ) 0 ( , ) 0 5 ,

1
| || || |

m n h k i j
h k h k i j

m n m n i j s t s t
h k i j s t s t

k p p p
r


  

    

( , ) ( , ) ( , )
,(1) , (2) , 4

1 2 , , ,

( , ) 0 ( , ) 0 ( , ) 0 , 3 5 ,

1
| || |

s tm n h k i j
i jh k h k

m n m n i j i j s t
h k i j s t s t s t

r
k k p p

r r




 



 
  

    

( , ) ( , ) ( , )
,, ,(1) , 2 4

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 1 , , 3 5 ,

1 2 3 , ,

1

| |

1
[ ; ]

i j s tm n h k i j
i jh k h kh k

m n m n h k i j s t
h k i j s t h k i j s t s t

m n m n

r r
k k p

r r r

k k k M p
r

 


   



 

  
  

  
 

where  

 2 1 1( 1)k t  ;  

3 2 2( 1)k t  . 

Therefore the Cannon works as follows 

(1)

6 1

1 1
[ ] [ ]
r r

    

chosen 1 6r and r  near to 0  we get  

  [0 ] 0   . 

That is to say, the similar transposed set  , ( , )m nu z w  was effective at origin. 

Now we are going to take into account non-algebraic sets of polynomials of two 

complex variables, for this suggestion we will take the following two lemmas: 

Lemma (2.3): 

If the transposed sets  ( )

, ( , ) ; 1,2i

m np z w i   satisfy condition (2.5) and the set 

 (1)

, ( , )m np z w  is general set satisfies the condition (2.6), which is effective at the origin 

of 
2C , then the similar transposed set  , ( , )m nu z w  holds: 

(2.20)   [0 ] 0   . 

Proof: 

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From effectiveness of the general set  (1)

, ( , )m np z w at the origin, that  

(2.21)  

1

(1) (1)

,

1
[0 ] sup [ ; ] 0

m n

m n
m n

Lim M p
r




 

 
   

 
. 

where   

 
1

(1) (1) , (1)

, , ,
, ( , )

[0 ] supsup ( , )

r

h k

m n m n h k
S h k

p p z w


  

 


    

or  

 (1)

,

3 4

1 1
[ ] ( ) ; , 0m n

m n k m n
r r

   . 

By conditions (2.5) and (2.6); we have  

(2.22)  
(2)

, ,

4 5

1 1
, ( )m n

m n m nM p k
r r

  
 
 

 

(2.23)  
(1)

, ,

3 2

1 1
, ( ) ; , 0m n

m n m nM p k m n
r r

  
 

 
. 

By (2.21), (2.22), (2.23) and Cauchy's inequality it follows that:  

(2.24)  
1

3

, ,

3

1
[ ; ] max ( , )

r

m n m n
S

M u u z w
r

  

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , ,

( , ) 0 ( , ) 0 ( , ) 0 3 ,

(1)

,( , ) ( , ) ( , )
(1) , (2) , (1) , 3

, , ,
(1)( , ) 0 ( , ) 0 ( , ) 0
,

3

1 1
| || || | ( )

1
[ , ]

| || || |
1

[ ,

m n h k i j
h k i j s t s t

m n h k i j

h k i j s t s t

i jm n h k i j
h k i j s t

m n h k i j

h k i j s t
i j

p p p
r

M P
r

p p p

M P
r





  

  





  

  
3 ,

1 1
( )

]

s t

s tr 



 

(1)

,( , ) ( , ) ( , )
(1) , (2) , 3

, ,
(1)( , ) 0 ( , ) 0 ( , ) 0 , 3

, ,

3

( , ) ( , ) ( , )
(1) , (2) , 2

1 , ,

( , ) 0 ( , ) 0 ( , ) 0 4 , 3

1
[ ]

1 1
| || | ( )

1
[ , ]

1 1
| || | ( )

i jm n h k i j
h k i j s t

m n h k

h k i j s t i j
s t s t

m n h k i j
h k i j s t

m n h k i j
h k i j s t i j

r
p p

r
M P

r

r
k p p

r r









  




  

   

  

 

( , ) ( , ) ( , )
(1) , (2) 2

1 , ,

( , ) 0 ( , ) 0 ( , ) 0 4 3

( , ) ( , )
(1) , 2

1 ,

( , ) 0 ( , ) 0 5 , 3

(1)

1 ,

5

1
| | [ , ]( )

1 1
| | ( )

1
[ , ].

m n h k i j
h k s t

m n h k

h k i j s t

m n i j
h k s t

m n h k
h k s t h k

m n

r
k p M P

r r

r
k p

r r

k M P
r





  




 

  

   

Taking  m n   and keeping in mind that the set  (1)

, ( , )m np z w  satisfies (2.5), we 

get   

   (1)0 0 0     . 

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Therefore we get  

   0 0   . 

 

 

Lemma (2.4): 

The similar transposed set  , ( , )m nu z w , satisfies the condition  

(2.25)    0 0r for r  

whenever the transposed sets  ( )

, ( , ) ; 1,2i

m np z w i   satisfy the condition (2.5) and 

(2.6) and the set  (2)

, ( , )m np z w  is effective at the origin. 

Proof: 

The transposed sets  ( )

, ( , ) ; 1,2i

m np z w i   satisfy the condition (2.6) we get  

(2.26)  (1)

, ,

4 5

1 1
[ ; ] ( )m n

m n h kM p k
r r

  . 

Also, the transposed set  (2)

, ( , )m np z w  is effective at the origin, satisfies conditions (2.5) 

and (2.6), so must be its inverse transposed set  (2)

, ( , )m np z w , thus we get   

(2.27)  (2)

, ,

3 4

1 1
[ ; ] ( )m n

m n m nM p k
r r

   

The similar transposed set  , ( , )m nu z w  written in the form:   

     (1) (2) (1)

, , , ,( , ) ( , ) ( , ) ( , )m n m n m n m nu z w p z w p z w p z w  

Therefore  

       (1) (1) (2)

, , , ,( , ) ( , ) ( , ) ( , )m n m n m n m np z w u z w p z w p z w  

from which we get   
( , ) ( , ) ( , )

(1) , (1) , (2) ,

, , , ,

( , ) 0 ( , ) 0 ( , ) 0

( , ) | || || |
m n h k i j

h k i j s t s t

m n m n h k i j

h k i j s t

p z w u p p z w
  

    . 

By (2.26), (2.27) and Cauchy's inequality we obtain: 
( , ) ( , ) ( , )

, (1) , (2) ,

, , , ,

( , ) 0 ( , ) 0 ( , ) 0 , 3

( , ) ( , ) ( , )
, (1) , (2) ,

, , , ,

( , ) 0 ( , ) 0 ( , ) 0 , 3

1
| || || |

1
| || || |

m n h k i j
m n h k i j s t

m n m n h k i j s t
h k i j s t s t

m n h k i j
h k i j s t

m n m n h k i j s t
h k i j s t s t

r u p p
r

u p p
r










  


  

  

  

 

( , ) ( , )
, (1) , (2)

, , , ,

( , ) 0 ( , ) 0 3

( , ) ( , )
, (1) ,

1 , , ,

( , ) 0 ( , ) 0 4 ,

1
| || | [ ; ]

1
| || |

m n h k
h k i j

m n m n h k i j

h k i j

m n h k
h k i j

m n m n h k i j
h k i j i j

u p M p
r

k u p
r






 


 

 

 
 

( , )
, (1)

1 , , ,

( , ) 0 4

( , )
,

1 , , 1 , ,

( , ) 0 5 , 5

1
| | [ ; ]

1 1
| | [ ; ]

m n
h k

m n m n h k

h k

m n
h k

m n m n m n m nh k
h k h k

k u M p
r

k u k M u
r r



 










 

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so that  

  
5

1
0r

r

 

 
 

. 

Chosen 5r  near to r  then we have  

    0 0r for r .  

Theorem (2.3): 

The general transposed set of polynomials of two complex variables   (1)

, ( , )m np z w  is 

effective at the origin of 
2C , satisfies condition (2.5) and (2.6). Then the similar 

transposed set  , ( , )m nu z w  is effective there, satisfies the conditions (2.20) and (2.25), 

if and only if, the transposed set  (2)

, ( , )m np z w  is effective at the origin of 
2C  and 

satisfies the same conditions. 

Proof: 

From effectiveness of general sets  ( )

, ( , ) ; 1,2i

m np z w i  at the origin, we get  

(2.28)  (1)

,

3 4

1 1
[ ] ( ) ; , 0m n

m n k m n
r r

   . 

(2.29)  (2)

,

4 5

1 1
[ ] ( ) ; , 0m n

m n k m n
r r

   . 

If the condition (2.20) of lemma (2.3) is satisfied then we get  

(2.30)  , ,

2 3

1 1
, ( )m n

m n m nM u k
r r

  
 
 

. 

By condition (2.6); we obtain   

(2.31)  
(1)

, ,

3 2

1 1
, ( ) ; , 0m n

m n m nM p k m n
r r

  
 

 
 

(2.32)  
(2)

, ,

4 3

1 1
, ( ) ; , 0m n

m n m nM p k m n
r r

  
 

 
. 

By (2.28) - (2.32) and Cauchy's inequality it follows that:  
( , )

,

, , , ,

( , ) 02 2

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , , , ,

( , ) 0 ( , ) 0 ( , ) 0 2

1 1
[ ] [ ; ]

1
| || || | [ ; ]

m n
h k

m n m n m n h k

h k

m n h k i j
h k i j s t

m n m n h k i j s t

h k i j s t

u M u
r r

p p p M u
r







  

 





  

 

( , ) ( , ) ( , )
(1) , (2) , (1) ,

1 , , , ,

( , ) 0 ( , ) 0 ( , ) 0 , 3

(1)

, ,( , ) ( , ) ( , )
(1) , (2) , (1) , 3

1 , , , ,

( , ) 0 ( , ) 0 ( , ) 0

1
| || || |

1
[ ; ]

| || || |

m n h k i j
h k i j s t

m n m n h k i j s t
h k i j s t s t

s t s tm n h k i j
h k i j s t

m n m n h k i j

h k i j s t

k p p p
r

M p
r

k p p p











  

  

  

  
(1) , 3

, ,

3

1

1
[ ; ]

s t

s t
s t s t

r
M p

r

 

 

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(1)

,( , ) ( , ) ( , )
(1) , (2) , 3 2

1 , , ,

( , ) 0 ( , ) 0 ( , ) 0 , 3

(2)

, ,( , ) ( , ) ( , )
(1) , (2) , 4

1 , , ,

( , ) 0 ( , ) 0 ( , ) 0
, ,

1
[ ]

| || |

1
[ ; ]

| || |

[

i j s tm n h k i j
h k i j

m n m n h k s t
h k i j s t s t

i j i jm n h k i j
h k i j

m n m n h k

h k i j s t
i j i

r r
k p p

r

M p
r

k p p

M p















  

  

  

   2

(2) 4 , 3

4

1

1
; ]

s t

i j s t

s t
j

r

r r

r





 

 

(2)

,( , ) ( , ) ( , )
(1) , 34 2

1 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 4 3

( , ) ( , ) ( , )
(1) , 3 2

1 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 5 4 3

1
[ ]

| |

1
| |

s ti jh km n h k i j
h k

m n m n

h k i j s t s t

s ti j
m n h k i j

h k

m n m n h k
h k i j s t h k

rr r
k p

r r

r r
k p

r r r











  




  

  
  

   

  
  

   

  

  

 

(1)

1 , ,

5

1
[ ; ]m n m nk M p

r
 . 

Therefore    

  [0 ] 0   . 

That to say the similar transposed set  , ( , )m nu z w effective at the origin, and satisfies 

conditions (2.20) and (2.25), by using lemmas (2) and (3) respectively and the "if" 

statement follows. 

Now, to prove the only if, write  

      (2) (1) (1)

, , , ,( , ) ( , ) ( , ) ( , )m n m n m n m np z w p z w u z w p z w  

let the set  , ( , )m nu z w  is effective there satisfies conditions (2.20) and (2.25), the 

inverse transposed set  (1)

, ( , )m np z w is effective at the origin and satisfies the same 

conditions. Then the similar transposed set  (2)

, ( , )m np z w  is effective at the origin, and 

satisfies conditions (2.20) and (2.25).  

3- Effectiveness of similar transposed set of polynomials in 

open hyperspheres   

Now we are investigating the effectiveness of similar transposed set  , ( , )m nu z w  of 

polynomials of two complex variables in open hyperspheres whenever the constituent 

sets are effective there. 

We can only be algebraic and compliance with the relevant requirements 

(3.1)    

1

( ) ( )

, ,

1 1 1
sup [ ; ] ; , 1,2

m n
i i

m n m n
m n

Lim M p for all r R i
r r R

 


 

   
   

   
. 

For this purpose, we give the following lemma: 

Lemma (3.1)  
The transposed set and the power transposed set accord to the same condition (3.1). 

Proof:  

First suppose that the set  , ( , )m np z w  satisfies condition  

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(3.2)  

1

, ,

1 1 1
sup [ ; ] ;

m n

m n m n
m n

Lim M p for all r R
r r R

 


 

   
   

   
. 

then  

(3.3)  
, ,

1

1 1
[ ; ] ( ) ; ( , ) 0m n

m n m nM p k m n
r r

   . 

The transposed product set of the two sets  ( )

, ( , ) ; 1,2i

m np z w i  , are satisfy the 

following relation: 

 
( , )

2 ,

, , ,

( , ) 0

( , ) ( , )
m n

h k

m n m n h k

h k

p z w p p z w


  .  

By relation (3.3) and using Cauchy's inequality we get   
( , )

2 ,

, , , , ,

( , ) 0

( , )
,

1 , ,

( , ) 0 1 ,

1 , ,

1

1 1
[ ; ] | | [ ; ]

1
| |

1
[ ; ].

m n
h k

m n m n m n m n h k

h k

m n
h k

m n m n h k
h k h k

m n m n

M p p M p
r r

k p
r

k M p
r

 











 

 . 

So that  

 2

2

1 1 1
[ ] [ ] ; .for all r R
r r R

    

Also by the same way we can prove that 

 1 2

2 3

1 1 1
[ ] [ ] ; for all r R
r r R

     

and by induction we get  

 
1 1

[ ] ; .for all r R
r R

  

We can deduce the following theorem from this Lemma: 

Theorem (3.1): 

When  ( )

, ( , ) ; 1,2i

m np z w i   two algebraic sets are effective and satisfy in the open 

hyperspheres 
1 rS  the condition (3.2), then the similar transposed set  , ( , )m nu z w  is 

effective in open hyperspheres 
1 RS . 

Proof: 

Let two sets  ( )

, ( , ) ; 1,2i

m np z w i   be algebraic sets each fulfilling the following 

conditions: 

 (3.4)  
,(1) , 3

, 1 1 1

, 2

( 1)
m n

h kh k

m n h k

m n

r
p k t

r









  

(3.5)   (2)

, , 1

2 3

1 1
[ ; ] ( )m n

m n i jM p k
r r

  . 

Therefore  

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4

, ,

4

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , ,

( , ) 0 ( , ) 0 ( , ) 0 4 ,

1
(3.6) [ , ] max ( , )

1
| || || |

r

m n m n
S

m n h k i j
h k i j s t

m n h k i j s t
h k i j s t s t

M u u z w
r

p p p
r 

  



   
 

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , ,

( , ) 0 ( , ) 0 ( , ) 0 4 ,

( , ) ( , ) ( , )
,(1) , (2) , 3

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 2 4 ,

1
| || || |

1
| || |

m n h k i j
h k i j i j

m n h k s t s t
h k i j s t s t

s tm n h k i j
i jh k i j

m n h k i j s t
h k i j s t s t s t

p p p
r

r
k k p p

r r





 


  



 
  

   

  

 

( , ) ( , ) ( , )
,(1) , (2) , 3

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 2 , 4 ,

( , ) ( , ) ( , )
, ,(1) , (2) 3

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 2 , 4 ,

1
| || |

1
| | [ , ]

s tm n h k i j
i jh k i j

m n h k i j s t
h k i j s t s t s t

s tm n h k i j
i j i jh k

m n h k s t
h k i j s t s t s t

r
k k p p

r r

r
k k p M p

r r



 

 

 



 
  




  

  

  

 

( , ) ( , ) ( , )
, ,(1) , 3

1 2 ,

( , ) 0 ( , ) 0 ( , ) 0 3 , , 4 ,

(1)

1 2 ,

3

1
| |

1
[ , ]

s tm n h k i j
i j i jh k

m n h k s t
h k i j s t h k s t s t

m n

r
k k p

r r

k k M p
r

 

  



 
  

  
 

where  

 2 1 1( 1)k t  . 

Thus  
1

, ,

4 3

1

(1) (1)

, 1 2 ,

3 3

1 1
[ ] limsup [ , ]

1 1
limsup [ ; ] [ ]

m n

m n m n
m n

m n

m n m n
m n

M u
r r

k k M p
r r

 

 



 



 

 
  

 

 
  

 

 

and  

 
, , 1

4 5

1 1
[ , ] ( )m n

m n m nM u k
r r

  .  

Now, taking 

(3.7)  
,(1) , 4

, 1 1 1

, 3

( 1)
m n

h kh k

m n h k

m n

r
p k t

r









  

(3.8)  
,(2) , 2

, 1 2 2

, 1

( 1)
m n

h kh k

m n h k

m n

r
p k t

r









 .  

By (3.6), (3.7), (3.8) and Cauchy's inequality it follows that:  
( , )

,

, , , ,

( , ) 04 4

( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , , , ,

( , ) 0 ( , ) 0 ( , ) 0 4

1 1
[ ] [ ; ]

1
| || || | [ ; ]

m n
h k

m n m n m n h k

h k

m n h k i j
h k i j s t

m n m n h k i j s t

h k i j s t

u M u
r r

p p p M u
r







  

 





  

 

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( , ) ( , ) ( , )
(1) , (2) , (1) ,

, , , , ,

( , ) 0 ( , ) 0 ( , ) 0 4

( , ) ( , ) ( , )
(1) , (2) , (1) ,

1 2 , , , ,

( , ) 0 ( , ) 0 ( , ) 0 5 ,

1
| || || | [ ; ]

1
| || || |

m n h k i j
h k h k i j

m n m n i j s t s t

h k i j s t

m n h k i j
h k h k i j

m n m n i j s t s t
h k i j s t s t

p p p M u
r

k k p p p
r






  


  

   

  
 

( , ) ( , ) ( , )
,(1) , (2) , 4

1 2 , , ,

( , ) 0 ( , ) 0 ( , ) 0 , 3 5 ,

( , ) ( , ) ( , )
,,(1) , 2 4

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 1 , 3

1
| || |

1
| |

s tm n h k i j
i jh k h k

m n m n i j i j s t
h k i j s t s t s t

i j s tm n h k i j
i jh kh k

m n m n h k i j
h k i j s t i j s t

r
k k p p

r r

r r
k k p

r r r




 




 



 
  

 

 
  

  

  
5 ,

s t

s t

 

( , ) ( , ) ( , )
,, ,(1) , 2 4

1 2 , ,

( , ) 0 ( , ) 0 ( , ) 0 , 1 , 1 , 3 5 ,

(1)

1 2 3 , ,

1

1
| |

1
[ ; ]

i j s tm n h k i j
i jh k h kh k

m n m n h k h k i j s t
h k i j s t h k i j s t s t

m n m n

r r
k k p

r r r r

k k k M p
r

 


   



 

   
  

  
 

where  

 2 1 1( 1)k t  ; 3 2 2( 1)k t  . 

So that  
1

(1)

,

4 4 1

1 1 1 1
[ ] limsup [ ] [ ] ;

m n

m n
m n

for all r R
r r r R

 


 

 
   

 
. 

In other words, the similar transposed set  , ( , )m nu z w  effective established an open 

hypersphere 
1 RS . 

4- Effectiveness of similar transposed set of polynomials in closed hyperspheres   

Now we are giving some important results for the effectiveness of similar transposed 

sets of polynomials in some other regions, and a new study of the effectiveness of these 

polynomials is being considered, and proof of these results is given in a similar manner 

to those previously identified in this study. 

First effectiveness of transposed basic set of polynomials of two complex variables 

 , ( , )m np z w  in closed hyperspheres 1

r

S whenever the simple basic set  , ( , )m np z w  

effective in closed hyperspheres 
rS  with leading coefficients unity.  

The effectiveness of transposed basic set of polynomials given in following theorem:  

Lemma (4.1): 

Suppose that , ( , )m np z w be simple monic set with leading coefficients unity effective 

in the hyperspheres 
rS , then the transposed set  , ( , )m np z w is effective in the closed 

hyperspheres 1

r

S . 

Let  , ( , )m np z w settheofpolynomialsofsetinversetransposedabe

 , ( , )m np z w where  

  
( , ) ( , )

, ,

, , ,

( , ) 0 ( , ) 0

( , )
m n m n

h k h k m n h k

m n m n h k

h k h k

p z w p z w p z w
 

   . 

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setinversetransposedtheEffectiveness  , ( , )m np z w in clospolynomialsof ed 

hyperspheres 1 ; 0
r

S r   with leading coefficients unity set, whenever the basic

 , ( , )m np z w  is effective in the same region under the same condition as follows: 

Lemma (4.2): 

Suppose that , ( , )m np z w be simple monic set with leading coefficients unity effective 

in the hyperspheres 
rS , then the transposed inverse set  , ( , )m np z w is effective in the 

closed hyperspheres 1

r

S . 

Let  ( )

, ( , ) ; 1,2i

m np z w i   are basic sets of polynomials of two complex variables and 

the set  , ( , )m nq z w is called the product set of the two sets  ( )

, ( , ) ; 1,2i

m np z w i  ,  

     (1) (2)

, , ,( , ) ( , ) ( , )m n m n m nq z w p z w p z w  , 

Now, the effectiveness of transposed product basic set of polynomials of two complex 

variables     (1) (2)

, , ,( , ) ( , ) ( , )m n m n m nq z w p z w p z w  in closed hyperspheres 1

r

S whenever 

the sets  ( )

, ( , ) ; 1,2i

m np z w i   closedineffectiveareones,are simple monic

hyperspheres 
rS  with leading coefficients unity, i.e. ( ) ,

, 1 ; 1,2i m n

m np i  . 

Lemma (4.3): 

Suppose that ( )

, ( , ) ; 1,2i

m np z w i   be simple monic set with leading coefficients unity 

effective in the hyperspheres 
rS transposedthethen, product set  , ( , )m nq z w is 

effective in the closed hyperspheres 1

r

S . 

effectiveness of similar transposed setNow, we give   , ( , )m nu z w  closedin

hyperspheres 1

r

S whenever the transposed simple basic sets  ( )

, ( , ) ; 1,2i

m np z w i   are 

hyperspheresclosedineffective 1

r

S  whenever the simple basic setsand also,

 ( )

, ( , )i

m np z w  are effective in open hyperspheres 
rS with leading coefficients unity, 

( ) ,

, 1i m n

m np  ; 1, 2i  . 

Theorem (4.1): 

Suppose that ( )

, ( , ) ; 1,2i

m np z w i   be two simple monic sets with leading coefficients 

unity effective in the hyperspheres 
rS , then the similar transposed set  , ( , )m nu z w is 

effective in the closed hyperspheres 1

r

S . 

The inverse set of polynomials of two complex variables  , ( , )m nu z w  as follows:  

       (1) (2) (1)

, , , ,( , ) ( , ) ( , ) ( , )m n m n m n m nu z w p z w p z w p z w  

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Now, we give effectiveness of inverse similar transposed set  , ( , )m nu z w  in closed 

hyperspheres 1

r

S , whenever the simple basic sets  ( )

, ( , ) ; 1,2i

m np z w i   are effective 

in closed  hyperspheres 
rS with leading coefficients unity, i.e. ( ) ,

, 1 ; 1,2i m n

m np i  . 

Theorem (4.2): 

Suppose that ( )

, ( , ) ; 1,2i

m np z w i   be two simple monic sets with leading coefficients 

in the hypersphereeffectiveunity
rS settransposedsimilarthen the inverse,

 , ( , )m nu z w  hypersphereclosedtheis effective in   1

r

S  settheifonlyandif

 (2)

, ( , )m np z w  is effective there. 

Conclusions 

In this paper, where the correctness of the corresponding functions is effectiveness in 

origin, in the open hyperspheres and in the closed hyperspheres, a new comparison is 

proposed to study some significant properties of some corresponding functions in two 

complex variables, and this study is called a new one of its kinds. Generalization of the 

corresponding position and it has relevance in many areas of application and physics. 

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