The Impact of Firm Size and Market Size Asymmetries on National Mergers in a Three-Country Model

Munich Personal RePEc Archive

The Impact of Firm Size and Market Size Asymmetries on National Mergers in a Three-Country Model

Santos-Pinto, Luís

University of Lausanne

26 August 2009

Online at https://mpra.ub.uni-muenchen.de/17166/

MPRA Paper No. 17166, posted 07 Sep 2009 19:12 UTC

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s ) % ( s( s ! 0 π^s_s1=

(a−c+ ∆)²/9 π^s_s2= (a−c−2∆)²/9. L(

q_L1= (a−c+ ∆)/3γ q_L2= (a−c−2∆)/3γ L ) ( L ( L π^L_L1= (a−c+ ∆)²/9γ π^L_L2= (a−c−2∆)²/9γ.

J 0 t ! 0 "

q_s1^t = a−c 2β −1

2 q^t_s2+q_L1^t +q^t_L2 q_s2^t = a−c−∆

2β −1

2 q_s1^t +q^t_L1+q^t_L2 q^t_L1 = a−c

2β −1

2 q^t_s1+q_s2^t +q_L2^t q^t_L2 = a−c−∆

2β −1

2 q_s1^t +q^t_s2+q_L1^t

! * 0 q_s1^t = q_L1^t = (a−c+ 2∆)/5β q_s2^t = q_L2^t = (a−c−3∆)/5β) ( s L ( t ! 0 π^t_s1 =π^t_L1 = (a−c+ 2∆)²/25β π^t_s2=π^t_L2= (a−c−3∆)²/25β)

s( s 0 J 0 J

qs+s = (a−c)/2. ( πs1+s2= (a−c)²/4. s (

L( % J 0 t ! 0

q^t_s1+s2 = a−c 2β −1

2 q^t_L1+q^t_L2 q^t_L1 = a−c

2β −1

2 q^t_s1+s2+q^t_L2 q^t_L2 = a−c−∆

2β −1

2 q_s1+s2^t +q_L1^t

! * 0 q^t_s1+s2=q^t_L1= (a−c+ ∆)/4β q^t_L2= (a−c− 3∆)/4β. ( s( t π^t_s1+s2= (a−c+ ∆)²/16β.

s( ( 0 * L ( (

3$

s( ( s( 0

% %

(a−c)²

4 +(a−c+ ∆)²

16β ≥ (a−c+ ∆)² 9 +(a−c−2∆)²

9 +(a−c+ 2∆)²

25β +(a−c−3∆)²

25β .

! J * β* 0 β≥f_s^L1,L2(δ)* !

+ ,) s( 0 L( % J 0 t !

0 "

q_s1^t = a−c 2β −1

2 q^t_s2+q_L1+L2^t q_s2^t = a−c−∆

2β −1

2 q_s1^t +q^t_L1+L2 q_L1+L2^t = a−c

2β −1

2 q^t_s1+q_s2^t

q^t_L1+L2 = q_s1^t = (a−c+ ∆)/4β q_s2^t = (a−c−3∆)/4β. ( s1 t π^t_s1 = (a−c+ ∆)²/16β

( s2 π^t_s2= (a−c−3∆)²/16β. s( L( *

! t) J 0 J t q_s1+s2^t =

q_L1+L2^t = (a−c)/3β ( s( 0 π^t_s1+s2= (a−c)²/9β.

% s( ( 0 * L(

(a−c)² 1 4+ 1

9β ≥(a−c+ ∆)² 1 9+ 1

16β +(a−c−2∆)²

9 +(a−c−3∆)²

16β .

! J * β* 0 β≥f_s^L1+L2(δ)* !

+ ,) % L( ( 0 * s(

(a−c)²

4γ +(a−c+ ∆)²

16β ≥ (a−c+ ∆)² 9γ +(a−c+ 2∆)²

25β +(a−c−2∆)²

9γ +(a−c−3∆)²

25β .

! J * β* 0 β≥f_L^s1,s2(δ, γ)* !

+ ,) L( ( 0 * s(

(a−c)² 1 4γ+ 1

9β ≥(a−c+ ∆)² 1 9γ+ 1

16β +(a−c−2∆)²

9γ +(a−c−3∆)²

16β .

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+ !,) 4 , 5

36

" + , H * 3 + , ( s

* * ( L * f_s^L1,L2(δ)≤β≤1.H *

+ , ( s* * ( L

* f_s^L1+L2(δ)≤β≤1. * ( s

! * ( L f_s^L1,L2(δ)−f_s^L1+L2(δ) =

1

10013+162δ−603δ

1+8δ−20δ >0 δ∈[0,1/3])

+ , ( L ) 3 + , * !

f_s^L1,L2(δ)≤0 δ∈[7/61,1/3]. δ∈[7/61,1/3]% (

s * * ( L β) 5 * ! %

δ∈[0,7/61)% ( s * * ( L

β≥f_s^L1,L2(δ))

+ , ( L ) 3 + , * !

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* ( L δ) 5 * ! % β∈(0,0.63)% ( s

* * ( L β≥f_s^L1,L2(δ))

+ !, H * 3 + , ( s* * (

L * f_s^L1,L2(δ)≤β≤1.H * 3 + ,

( L* * ( s * f_L^s1,s2(δ, γ)≤

β≤1. * ( L * ( s

! * ( s

* ( L f_L^s1,s2(δ, γ)≤f_s^L1,L2(δ)

(γ, δ)) 4 , 5

0 . 0 ( )

( ( s

+ , ( * ( s+

,) * ( (

L+ , ( * ( L +

,) < ( C m + ,

n+ , M + , N + ,)

< 0 ( 0 F2,γ J 0 0 6,(F2,γ))

0

s L M N

m

(a−c) 4

+^(a−c)_9β ,

(a−c) 4γ

+^(a−c)_9β

(a−c) 4

+^(a−c+∆)_16β ,

(a−c+∆) +(a−c−2∆) 9γ

+(a−c+∆) +(a−c−3∆) 16β

n

(a−c+∆) +(a−c−2∆) 9

+(a−c+∆) +(a−c−3∆)

16β ,

(a−c) 4γ

+^(a−c+∆)_16β

(a−c+∆) +(a−c−2∆) 9

+(a−c+2∆) +(a−c−3∆)

25β ,

(a−c+∆) +(a−c−2∆) 9γ

+(a−c+2∆) +(a−c−3∆) 25β

3A

- " . δ^∗(γ) = _750−549γ^50−63γ .

" (γ, δ) 0 < γ < 0.794 0 ≤ δ ≤ δ^∗(γ), f_L^s1+s2(δ, γ) <

f_L^s1,s2(δ, γ)< f_s^L1+L2(δ)< f_s^L1,L2(δ).

" δ=δ^∗(γ), f_L^s1+s2(δ, γ)< f_L^s1,s2(δ, γ) =f_s^L1+L2(δ)< f_s^L1,L2(δ).

" (γ, δ) 0< γ ≤0.794 δ^∗(γ)< δ≤0.0(6), f_L^s1+s2(δ, γ)<

f_s^L1+L2(δ)< f_L^s1,s2(δ, γ)< f_s^L1,L2(δ).

" 0.794< γ <1, f_L^s1+s2(δ, γ)< f_s^L1+L2(δ)< f_L^s1,s2(δ, γ)< f_s^L1,L2(δ).

" γ= 1, f_L^s1+s2(δ, γ) =f_s^L1+L2(δ)< f_L^s1,s2(δ, γ) =f_s^L1,L2(δ).

- " δ^∗(γ) 0 0 ! f_s^L1+L2(δ) = f_L^s1,s2(δ, γ)

* δ) *% 0 < γ < 0.794 0 ≤ δ^∗(γ) ≤ 0.0(6). %

0 < γ < 0.794 0 ≤ δ ≤ δ^∗(γ), f_L^s1,s2(δ, γ) < f_s^L1+L2(δ). 5 * ! % 0 < γ ≤0.794 δ^∗(γ) < δ ≤ 0.0(6), f_s^L1+L2(δ) < f_L^s1,s2(δ, γ).

( f_L^s1+s2(δ, γ) f_s^L1+L2(δ) f_L^s1+s2(δ, γ) < f_s^L1+L2(δ) γ∈(0,1) f_L^s1+s2(δ, γ) =f_s^L1+L2(δ)* γ= 1. % ( f_L^s1,s2(δ, γ) f_s^L1,L2(δ) f_L^s1,s2(δ, γ) < f_s^L1,L2(δ) γ ∈ (0,1)

f_L^s1+s2(δ, γ) =f_s^L1+L2(δ)* γ= 1. + , +!,) 4 , 5

" + , 0.74≤γ≤1 β≤f_s^L1+L2(δ),

# + !, β ≤ f_s^L1+L2(δ) < f_s^L1,L2(δ)) ) 3 + , + , *

β ≤ f_s^L1+L2(δ) < f_s^L1,L2(δ) m (

s) % ( s n) β ≤ f_s^L1+L2(δ), # + !,

β < f_L^s1,s2(δ, γ). ) 3 + , * β < f_L^s1,s2(δ, γ)

0 ( L n N) % ( L * N) %

6,(F_2,γ) = (n, N) 0.794≤γ≤1 β≤f_s^L1+L2(δ).

+ , 0.794 ≤ γ ≤ 1 f_s^L1+L2(δ) < β ≤ f_L^s1,s2(δ, γ), # + !, f_L^s1+s2(δ, γ)< f_s^L1+L2(δ)< β≤f_L^s1,s2(δ, γ)< f_s^L1,L2(δ). f_s^L1+L2(δ)<

β < f_s^L1,L2(δ)% ) 3 + , 0 ( s N

n ) 3 + , 0 ( s M m)

f_L^s1+s2(δ, γ)< β≤f_L^s1,s2(δ, γ)% ) 3 + , 0

( L n N ) 3 + !, 0 (

L m M) %(n, N) (m, M) J 0 + -

* , F_2,γ * 0.794 ≤γ ≤1 f_s^L1+L2(δ)}< β ≤f_L^s1,s2(δ, γ))

/ / J 0 + - * , *

( s 0 * m n ( L . 0 *

M N"

ps

(a−c)²

4γ +(a−c)²

9β + (1−ps) (a−c)²

4γ +(a−c+ ∆)² 16β

=ps

(a−c+ ∆)²

9γ +(a−c+ ∆)²

16β +(a−c−2∆)²

9γ +(a−c−3∆)² 16β +(1−ps) (a−c+ ∆)²

9γ +(a−c+ 2∆)²

25β +(a−c−2∆)²

9γ +(a−c−3∆)²

25β ,

3'

* ps 0 0 ( s m) ! J ps

* 0 +3,) L 0 * M N ( s

. 0 * m n) pL 0 0 ( L

M) γ= 1 +3, * 0 +#,) % 0.794≤γ≤1 f_s^L1+L2(δ)<

β≤f_L^s1,s2(δ, γ),* ! 6,(F2,γ) ={(n, N),(m, M),(ps, m;pL, M)}.

+ , 0.794≤γ≤1 f_L^s1,s2(δ, γ)< β, # + !, f_L^s1+s2(δ, γ)

< f_L^s1,s2(δ, γ)< β) ) 3 + , + !, * f_L^s1+s2(δ, γ)< f_L^s1,s2(δ, γ)

< β N ( L) % ( L

M) f_L^s1,s2(δ, γ)< β, # + !, f_s^L1+L2(δ)< β. ) 3

+ , * f_s^L1+L2(δ)< β 0 ( s

M m) % ( s m) % 0.794≤γ≤1 β≤f_s^L1+L2(δ)

* ! 6,(F2,γ) = (m, M). 4 , 5

" + , 0 < γ < 0.794 β ≤ min[f_s^L1+L2(δ), f_L^s1,s2(δ, γ)], # + , + , β≤f_s^L1+L2(δ)< f_s^L1,L2(δ)) ) 3 + , + , * β ≤f_s^L1+L2(δ)< f_s^L1,L2(δ) m

( s) % ( s n) β≤f_L^s1,s2(δ, γ),

) 3 + , 0 ( L n N) % (

L* N) % 0< γ <0.794 β≤min{f_s^L1+L2(δ), f_L^s1,s2(δ, γ)}

* ! 6,(F2,γ) = (n, N).

+ , 0< γ <0.794 f_L^s1,s2(δ, γ)≤β≤f_s^L1+L2(δ)% # + , f_L^s1+s2(δ, γ)< f_L^s1,s2(δ, γ)≤β≤f_s^L1+L2(δ)< f_s^L1,L2(δ). ) 3 + , + !, * f_L^s1+s2(δ, γ) < f_L^s1,s2(δ, γ) ≤ β N

( L) % ( L M) ) 3 + ,

+ , * β≤f_s^L1+L2(δ)< f_s^L1,L2(δ) m

( s) % ( s n) % 0 < γ < 0.794

f_L^s1,s2(δ, γ) ≤ β ≤ f_s^L1+L2(δ) * ! 6,(F2,γ) = (n, M).

+ , + !, + , + , ) #%

! ) 4 , 5

- "

" 0< β ≤g_s^L1,L2(δ) = ₅₀⁹ −7+82δ−183δ 5−32δ+44δ ,

#

" 0 < β ≤ g_s^L1+L2(δ) = ^{−1+18δ−45δ}_{5−32δ+44δ} ,

#

" 0< β ≤g^s1,s2_L (δ, γ) = ^9γ₅₀−7+82δ−183δ 5−32δ+44δ ,

#

" 0< β≤g_L^s1+s2(δ, γ) =γ^{−1+18δ−45δ}_{5−32δ+44δ}

- " 0 *

! * ! L)

s ! 0 CSs= (a−ps)Qs/2 =Q²_s/2,* Qs

0 s( ) s( % Qs= (2a−2c−∆)/3

#&

CS_s1,s2^s = (2a−2c−∆)²/18. s ( % Qs = (a−c)/2

CS^s_s1+s2 = (a−c)²/8) % ( s ! *

* L(

(a−c)²

8 +(a−c)²

4 +(a−c+ ∆)²

16β ≥ (2a−2c−∆)² 18 +(a−c+ ∆)²

9 +(a−c+ 2∆)²

25β +(a−c−2∆)²

9 +(a−c−3∆)² 25β . +?,

! +?, * β * 0 β ≤ g_s^L1,L2(δ) * ! + ,)

( s ! * * L(

(a−c)² 1 8+1

4+ 1

9β ≥ (2a−2c−∆)² 18 +(a−c+ ∆)²

9 +(a−c+ ∆)²

16β +(a−c−2∆)²

9 +(a−c−3∆)²

16β . +$,

! +$, * β* 0 β≤g_s^L1+L2(δ)* ! + ,) *

* * L ! L *

! s) L ! 0 CSL =

(a−pL)QL/2 =γQ²_L/2,* QL 0 L( ) L(

% QL= (2a−2c−∆)/3γ CS_L1,L2^L = (2a−2c−∆)²/18γ.

L ( % QL = (a−c)/2γ CS_L1+L2^L = (a−c)²/8γ) %

( L ! * * s(

(a−c)²

8γ +(a−c)²

4γ +(a−c+ ∆)²

16β ≥ (2a−2c−∆)² 18γ +(a−c+ ∆)²

9γ +(a−c+ 2∆)²

25β +(a−c−2∆)²

9γ +(a−c−3∆)² 25β . +6,

! +6, * β * ! β ≤ g_L^s1,s2(δ, γ) * ! + ,)

( L ! * * s(

(a−c)² 1 8γ + 1

4γ + 1

9β ≥(2a−2c−∆)² 18γ +(a−c+ ∆)²

9γ +(a−c+ ∆)²

16β +(a−c−2∆)²

9γ +(a−c−3∆)²

16β . +A,

! +A, * β* ! β≤g_L^s1+s2(δ, γ)* ! + !,) 4 , 5

" * 4 ;

) 3 * 3) 4 , 5

0 . 0 !

)

#3

( ( s+ , * (

* ( s+ ,) *

( ( L +

, * ( * ( L+ ,) <

! C m+ , n + ,

M+ , N + ,) <

0 ! 0 G_2,γ J 0 0 6,(G_2,γ))

0

s\L M N

m

(a−c) 8

+^(a−c)₄ +^(a−c)_9β ,

(a−c) 8γ

+^(a−c)_4γ +^(a−c)_9β

(a−c) 8

+^(a−c)₄ +^(a−c+∆)_16β ,

(2a−2c−∆) 18γ

+(a−c+∆) +(a−c−2∆) 9γ

+(a−c+∆) +(a−c−3∆) 16β

n

(2a−2c−∆) 18

+(a−c+∆) +(a−c−2∆) 9

+(a−c+∆) +(a−c−3∆)

16β ,

(a−c) 8γ

+^(a−c)_4γ +^(a−c+∆)_16β

(2a−2c−∆) 18

+(a−c+∆) +(a−c−2∆) 9

+(a−c+2∆) +(a−c−3∆)

25β ,

(2a−2c−∆) 18γ

+(a−c+∆) +(a−c−2∆) 9γ

+(a−c+2∆) +(a−c−3∆) 25β

- . ˆδ(γ) = _549−750γ^63−50γ .

" (γ, δ) 0< γ ≤0.466 0.11475 < δ≤ˆδ(γ), g^s1,s2_L (δ, γ)<

g^L1,L2_s (δ)< g^s1+s2_L (δ, γ)< g^L1+L2_s (δ).

" δ= ˆδ(γ) g_L^s1,s2(δ, γ)< g_s^L1,L2(δ) =g_L^s1+s2(δ, γ)< g_s^L1+L2(δ).

" (γ, δ) 0< γ≤0.466 ˆδ(γ)≤δ≤0.1991, g^s1,s2_L (δ, γ)<

g^s1+s2_L (δ, γ)< g^L1,L2_s (δ)< g^L1+L2_s (δ).

" 0.466< γ <1, g^s1,s2_L (δ, γ)< g^L1,L2_s (δ)< g^s1+s2_L (δ, γ)< g_s^L1+L2(δ).

" γ= 1, g_L^s1,s2(δ, γ) =g_s^L1,L2(δ)< g_L^s1+s2(δ, γ) =g_s^L1+L2(δ).

- " / ˆδ(γ) 0 0 ! g^L1,L2_s (δ) =

g^s1+s2_L (δ, γ)* δ) *% 0< γ ≤ 0.466 0.11475< ˆδ(γ) ≤ 0.1991. % 0 < γ ≤ 0.466 ˆδ(γ) ≤ δ ≤ 0.1991% g^s1+s2_L (δ, γ) <

g^L1,L2_s (δ).5 * ! % 0< γ≤0.466 0.11475< δ≤ˆδ(γ), g_s^L1,L2(δ)<

g^s1+s2_L (δ, γ)) ( g^s1+s2_L (δ, γ) g_s^L1+L2(δ) g_L^s1+s2(δ, γ)

< g_s^L1+L2(δ) γ∈(0,1) g^s1+s2_L (δ, γ) =g^L1+L2_s (δ)* γ= 1. % ( g^s1,s2_L (δ, γ) g^L1,L2_s (δ) g_L^s1,s2(δ, γ) < g_s^L1,L2(δ) γ∈(0,1) g^s1+s2_L (δ, γ) =g^L1+L2_s (δ)* γ= 1. *

* + , +!,) 4 , 5

##

!" + , 0.466 ≤ γ ≤ 1 g^s1+s2_L (δ, γ) < β, 7 + !, g_s^L1,L2(δ)< g_s^L1+L2(δ)< β) ) 7 + , + ,

* g_s^L1,L2(δ)< g_s^L1+L2(δ)< β m

! s) % ! s n) g_L^s1+s2(δ, γ)< β,

7 + !, g_L^s1,s2(δ, γ)< β. ) 7 + ,

* g_L^s1,s2(δ, γ) < β 0 ! L

n N) % ! L N) % 0.466 ≤ γ ≤ 1

g^s1+s2_L (δ, γ)< β * ! 6,(G_2,γ) = (n, N).

+ , 0.466≤γ≤1 g_s^L1,L2(δ)< β≤g^s1+s2_L (δ, γ), 7 + !, g_L^s1,s2(δ, γ)< g^L1,L2_s (δ)< β ≤g_L^s1+s2(δ, γ)< g^L1+L2_s (δ). g^L1,L2_s (δ)<

β < g_s^L1+L2(δ)% ) 7 + , 0

! s N n ) 7 + , 0

! s M m) g^s1,s2_L (δ, γ)< β≤g^s1+s2_L (δ, γ)% ) 7

+ , 0 ! L n N

) 7 + !, 0 ! L m

M) % (n, N) (m, M) - G_2,γ * 0.466 ≤ γ ≤1 g^L1,L2_s (δ)< β≤g^s1+s2_L (δ, γ)) / - * !

s 0 * m n ! L .

0 * M N"

q_s(a−c)² 1 8γ+ 1

4γ + 1

9β + (1−q_s) (a−c)²

8γ +(a−c)²

4γ +(a−c+ ∆)² 16β

=qs

(2a−2c−∆)²

18γ +(a−c+ ∆)²+ (a−c−2∆)² 9γ

+qs

(a−c+ ∆)²+ (a−c−3∆)²

16β + (1−qs)(2a−2c−∆)² 18γ +(1−qs) (a−c+ ∆)²+ (a−c−2∆)²

9γ +(a−c+ 2∆)²+ (a−c−3∆)²

25β ,

* qs 0 0 ! s m) !

J qs* 0 +4,) ! L 0 * M

N ! s . 0 * m n) qL

0 0 ! L M) γ = 1 +4, *

0 +7,) % 0.466≤γ≤1 g_s^L1,L2(δ)< β≤g_L^s1+s2(δ, γ), * ! 6,(G_2,γ) ={(n, N),(m, M),(qs, m;qL, M)}.

+ , 0.466 ≤ γ ≤ 1 β ≤ g_s^L1,L2(δ), 7 + !,

β < g_s^L1,L2(δ) < g_s^L1+L2(δ)) ) 7 + , + !, * β <

g^L1,L2_s (δ)< g^L1+L2_s (δ) n !

s) % ! s m) β ≤ g_s^L1,L2(δ),

7 + !, β < g^s1+s2_L (δ, γ). ) 7 + !, * β <

g^s1+s2_L (δ, γ) 0 ! L m M)

% ! L M) %6,(G2,γ) = (m, M) 0.466≤γ≤1

β≤g_s^L1,L2(δ). 4 , 5

#4

#" + , 0< γ <0.466 max{g_L^s1+s2(δ, γ), g^L1,L2_s (δ)}

< β, 7 + , + , g_L^s1,s2(δ, γ) < g_L^s1+s2(δ, γ) < β) ) 7 + , + !, * g_L^s1,s2(δ, γ) < g^s1+s2_L (δ, γ) < β M

! L) g_s^L1,L2(δ) < β, ) 7

+ , 0 ! s N n) %

6,(G_2,γ) = (n, N) 0.466≤γ≤1 max{g_L^s1+s2(δ, γ), g^L1,L2_s (δ)}< β.

+ , 0 < γ < 0.466 g_L^s1+s2(δ, γ) ≤ β ≤ g_s^L1,L2(δ)% 7 + , g^s1,s2_L (δ, γ) < g_L^s1+s2(δ, γ) ≤ β ≤ g^L1,L2_s (δ) < g^L1+L2_s (δ). ) 7 + , + !, * g^s1,s2_L (δ, γ) < g^s1+s2_L (δ, γ) ≤ β M

! L) ) 7 + , + ,

* β ≤ g^L1,L2_s (δ) < g^L1+L2_s (δ) n

! s) % 6,(G2,γ) = (m, N) 0 < γ < 0.466 g^s1+s2_L (δ, γ)≤β≤g^L1,L2_s (δ). + , + !,

+ , + , ) ?) 4 , 5

%" . 0.794≤γ≤1.

+ ," 0< β≤f_s^L1+L2(δ) NE(F_2,γ) =N E(G_2,γ) = (n, N)#

+ , " max{f_L^s1,s2(δ, γ), g^s1+s2_L (δ, γ)} ≤ β ≤ 1 N E(F2,γ) = (m, M) =

(n, N) =N E(G2,γ)#

+ ," 0< β≤min{g_s^L1,L2(δ),1} N E(F2,γ) =N E(G2,γ) = (m, M)

%" * # ?)4 , 5

'" . 0.466≤γ <0.794.

+ , " 0 < β ≤ min{f_s^L1+L2(δ), f_L^s1,s2(δ, γ)} N E(F_2,γ) = N E(G_2,γ) = (n, N)#

+ , " f_L^s1,s2(δ, γ) ≤ β ≤ f_s^L1+L2(δ) NE(F_2,γ) = (n, M) = (n, N) =

NE(G2,γ)#

+ ," max {f_s^L1+L2(δ), f_L^s1,s2(δ, γ), g_L^s1+s2(δ, γ)} ≤β≤1 N E(F_2,γ) = (m, M) = (n, N) =N E(G_2,γ)#

+ ," 0< β≤min{g_s^L1,L2(δ),1} NE(F_2,γ) =NE(G_2,γ) = (m, M)

'" * 4 ?)4 , 5

*" . 0< γ <0.466.

+ , " 0 < β ≤ min{f_s^L1+L2(δ), f_L^s1,s2(δ, γ)} N E(F2,γ) = N E(G2,γ) = (n, N)#

+ , " f_L^s1,s2(δ, γ) ≤ β ≤ f_s^L1+L2(δ) NE(F2,γ) = (n, M) = (n, N) =

NE(G2,γ)#

+ , " max{f_s^L1+L2(δ), f_L^s1,s2(δ, γ), g_L^s1+s2(δ, γ), g^L1,L2_s (δ)} ≤ β ≤ 1 NE(F_2,γ) = (m, M) = (n, N) =NE(G_2,γ)#

+ , " g_L^s1+s2(δ, γ) < β ≤ g^L1,L2_s (δ) N E(F_2,γ) = (m, M) = (m, N) =

NE(G_2,γ)#

+ ," 0< β ≤min{g^L1,L2_s (δ), g_L^s1+s2(δ, γ),1} NE(F_2,γ) =N E(G_2,γ) = (m, M)

*" * 4 $)4 , 5

#7