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f ( px + (1 p)y ) + f ( (1 p)x + py ) = f (x)+ f (y), (x, y I ) 0 <p< 1 f : I R p 0 <p< 1 f ( px + (1 p)y ) + f ( (1 p)x + py ) = f (x)+ f (y), p

Functional equations involving means

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Page 1: Functional equations involving means

f(px + (1 − p)y

)+ f

((1 − p)x + py

)= f(x) + f(y), (x, y ∈ I)

0 < p < 1 f : I → R

p

0 < p < 1

f(px + (1− p)y

)+ f

((1− p)x + py

)= f(x) + f(y),

p

©

Page 2: Functional equations involving means

f : I → R I ⊂ Rx, y ∈ I Sp(I) f : I → R

Sp(I) = S1−p(I)

f ∈ Sp(I) k

Ak : Rk → R k = 0, 1, 2

A2

(px, (1− p)x

)= 0 (x ∈ R)

f(x) = A2(x, x) + A1(x) + A0

x ∈ I

pp ∈ ]0, 1[ f : I → R p I

f(px + (1− p)y

)+ f

((1− p)x + py

)� f(x) + f(y)

x, y ∈ I f : I → R −f : I → R pI f ∈ Sp(I)

p I

f ∈ Sp(I) ξ ∈ Iδ > 0 ]ξ − δ, ξ + δ[ ⊂ I f | ]ξ − δ, ξ + δ[

Sp

(]ξ − δ, ξ + δ[

)f : I → R ξ ∈ I

I ξ f fI

0 < p < 1A2

p

f(αx + (1− α)y

)+ f

((1− α)x + αy

)= f

(βx + (1− β)y

)+ f

((1− β)x + βy

)x, y ∈ I f : I → R 0 < α < 1 0 < β < 1

α �∈ {β, 1− β}

Page 3: Functional equations involving means

p

A2 : R2 → Rα

A2(αx, x) = 0 x ∈ R,

−α α−α

α α∑n

i=1 tiαi = 0 ti ∈ Z i = 0, . . . , n

tn �= 0

A2(αkx, y) = (−1)kA2(x, αky)

k � 0 x, y ∈ R k = 0

0 = A2

(α(x− y), x− y

)= A2(αx, x)−A2(αy, x)−A2(αx, y) + A2(αy, y)

= −A2(αy, x)−A2(αx, y)

x, y ∈ R A2(αx, y) = (−1)A2(x, αy)k = 1 k � 1

A2(αk+1x, y) = A2(ααkx, y) = −A2(αkx, αy)

= (−1)(−1)kA2(x, αkαy) = (−1)k+1A2(x, αk+1y),

k + 1

β :=n∑

i=0

ti(−α)i �= 0.

x, y ∈ R z :=y

β

0 = A2(0, z) = A2

(( n∑i=0

tiαi

)x, z

)=

n∑i=0

tiA2(αix, z)

=n∑

i=0

ti(−1)iA2(x, αiz) = A2

(x,

( n∑i=0

ti(−1)iαi

)z

)= A2(x, βz) = A2(x, y),

A2 �

Page 4: Functional equations involving means

α −αα

A2 : R2 → R α α−α α

α βa : R → R

a(αx) = βa(x) x ∈ R.

α βα a : R → R

a(αx) = βa(x) x ∈ R.

α α −αa : R → R

a(αx) = −αa(x) x ∈ R.

a : R → R

A2(x, y) := a(x)y + a(y)x x, y ∈ R.

A2 : R2 → R

A2(αx, x) = a(αx)x + a(x)αx = −αa(x)x + a(x)αx = 0

x ∈ R �

0 < p < 1 f ∈ Sp(I)

2f

(x + y

2

)= f(x) + f(y) (x, y ∈ I)?

Page 5: Functional equations involving means

0 < p < 11−p

p −1−pp

f ∈ Sp(I)1−p

p1−p

p

−1−pp f ∈ Sp(I)

A2 : R2 → Rpx = y

A2

(1− p

py, y

)= 0 y ∈ R

A21−p

p −1−pp

A2 : R2 → R

A2

(1− p

px, x

)= 0 x ∈ R,

x �→ A(x, x) x ∈ I

pp

p pp p

p 1−pp −1−p

p

p ∈ ]0, 1[ p �= 12

1−pp −1−p

p

p p2p−1

α β

0 < α < β � 12.

Page 6: Functional equations involving means

a < b a, b ∈ I Pa,b(I)(a, a)

(αa + (1− α)b, (1− α)a

+ αb) (

(1− α)a + αb, αa + (1− α)b)

(b, b)

P (I) :=⋃

a,b∈I, a<b

Pa,b(I).

f : I → Rx, y ∈ I p :=

α+β−12α−1 ∈ ]0, 1[ f : I → R

f(pu + (1− p)v

)+ f

((1− p)u + pv

)= f(u) + f(v)

(u, v) ∈ P (I) ⊂ I2 P (I) R2

{u = αx + (1− α)y

v = (1− α)x + αy(x, y) ∈ I2,

(x, y) ∈ I2 (u, v) ∈ I2

{(u, v) =(αx + (1− α)y, (1− α)x + αy

)| x, y ∈ I} = P (I),

P (I)x y ⎧⎪⎪⎨⎪⎪⎩

x =α

2α− 1u +

α− 12α− 1

v

y =α− 12α− 1

u +α

2α− 1v

(u, v) ∈ P (I).

f(u) + f(v)

= f

2α− 1u +

α− 12α− 1

v

)+ (1− β)

(α− 12α− 1

u +α

2α− 1v

))+ f

((1− β)

2α− 1u +

α− 12α− 1

v

)+ β

(α− 12α− 1

u +α

2α− 1v

))= f

(pu + (1− p)v

)+ f

((1− p)u + pv

),

Page 7: Functional equations involving means

f : I → Rx, y ∈ I f ∈ Sp(I) p := α+β−1

2α−1 ∈ ]0, 1[

(u, v) ∈ P (I) ⊂ I2

ξ ∈ I δ > 0]ξ − δ, ξ + δ[ ⊂ I f | ]ξ − δ, ξ + δ[ ∈ Sp

(]ξ − δ, ξ + δ[

)I2 :=

{(ξ, ξ) | ξ ∈ I

}P (I)

δ > 0 ξ ∈ I ]ξ − δ, ξ + δ[2 ⊂ P (I)f | ]ξ − δ, ξ + δ[ ∈ Sp

(]ξ − δ, ξ + δ[

)�

α, β ∈ ]0, 1[ α �∈ {β, 1− β} A2 : R2 → R

A2

(αx, (1− α)x

)= A2

(βx, (1− β)x

)(x ∈ R)

A2

(α + β − 1

α− βx, x

)= 0 (x ∈ R).

A2

((α + β − 1)y, (α− β)y

)= A2

(βy, (1− β)y

)−A2

(αy, (1− α)y

)y ∈ R

A2

((α + β − 1)y, (α− β)y

)= 0 (y ∈ R).

x = (α− β)y�

α,β ∈ ]0, 1[ α �∈ {β,1−β} f : I →R

f(αx + (1− α)y

)+ f

((1− α)x + αy

)= f

(βx + (1− β)y

)+ f

((1− β)x + βy

)(x, y ∈ I) k Ak : Rk

→ R (k = 0, 1, 2)

A2

(αx, (1− α)x

)= A2

(βx, (1− β)x

)(x ∈ R)

f(x) = A2(x, x) + A1(x) + A0 x ∈ I.

Page 8: Functional equations involving means

f : I → Rf ∈ Sp(I) p = α+β−1

2α−1 ∈ ]0, 1[k Ak : Rk

→ R (k = 0, 1, 2)

A2

(px, (1− p)x

)= 0 (x ∈ R)

A2

(p

1− py, y

)= 0 (y ∈ R),

A2

(α + β − 1

2α− 1· 2α− 1

α− βy, y

)= 0 (y ∈ R).

�f

x �→ A2(x, x) x ∈ I

α, β ∈ ]0, 1[ α �∈ {β, 1−β}A2 : R2 →R

α+β−1α−β −α+β−1

α−β

x �→ A2(x, x) x ∈ I

t :=α + β − 1

α− β− t t.

A(α, β)A2 : R2 → R α, β ∈ ]0, 1[ α �∈ {β, 1− β}

α β12

α+β−1α−β A(α, β) �= ∅

α β 12 A(α,β) = ∅

Page 9: Functional equations involving means

a ∈ {2√

2, 2 3√

2}

α := limn→∞

(1− 1

n

)n

=1e, β :=

a

a− 1− a + 1

a− 11e,

α, β ∈ ]0, 1[ α �∈ {β, 1− β} α+β−1β−α = 1

a A(α, β) �= ∅a = 2

√2 A(α, β) = ∅ a = 2 3

√2

t