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Tan et al. Complex Eng Syst 2023;3:6 I http://dx.doi.org/10.20517/ces.2023.10 Page 11 of 23

According to Lemma 1, one further comes to

∫ T T T T T T ˜ −1
˜ −1

− ¤ ( )R ¤ ( ) ≤ ( )Ξ ( ) + (1 − ) S R S 1 + S R S 2 , (24)

2
1
−
where
[ ]
T T T T T T
( ) = ( ) ( − ( )) ( − ) 2 1 2 2 ,

 11 
Ξ
 ∗ ∗ ∗ ∗ 
 21 22 
 Ξ Ξ ∗ ∗ ∗ 
 31 32 33 
Ξ = Ξ Ξ Ξ ∗ ∗  ,
 41 42 43 44 
Ξ Ξ Ξ Ξ ∗ 
 
 51 Ξ 52 Ξ 53 Ξ 54 Ξ 55 
Ξ
 
Ξ 11 = −4(1 + )R , Ξ 21 = −2(1 + )R − S 1 − S 2 − S 3 − S 4 ,

Ξ 22 = −2(4 + )R − 2S 1 + 2S 2 − 2S 3 + 2S 4 , Ξ 31 = −S 1 − S 2 + S 3 + S 4 ,

Ξ 32 = −2(2 − )R + S 1 − S 2 − S 3 + S 4 , Ξ 33 = −4(2 − )R , Ξ 41 = −S 3 − S 4 ,

44
Ξ 42 = S 3 − S 4 + 3(2 − )R , Ξ 43 = 3(2 − )R , Ξ = −3(2 − )R ,

51 52 53 54 55
Ξ = 3(1 + )R , Ξ = −S 2 + 3(1 + )R , Ξ = −S 4 , Ξ = S 4 , Ξ = −3(1 + )R .
And
T

¤ ( )R ¤ ( )

∑ ∑
( ) T
= (A ( ) + B ( − ( )) + B ( ))

=1 =1

∑ ∑
( )
R (A ( ) + B ( − ( )) + B ( ))

=1 =1

∑ ∑ ∑
T T T T T
+2 ( )A R G ( ) + 2 ( − ( )) B R G ( )

=1, ≠ =1 =1, ≠

∑ T ∑ ∑ ∑
T
T
T
+2 ( ( ) ( )) B R G ( ) + ( )G T R G ( ),

=1 =1, ≠ =1, ≠ =1, ≠