Tincu Ioan. 199KB Jun 04 2011 12:09:37 AM

SOME SUMABILITY METHODS

by

Ioan Tincu

Let A =‖ρk(n)‖n N, k=0, …, n be a matrix with ρk(n) R. A sequence s = {sn}n N is

said to be A-summable to ρ if

∑

ρ ⋅ =ρ

= ∞ →

n

k

k k

n n s

0

) (

lim .

A sequence a = (an)n≥0 is called p, q-convex if

an+2 – (p + q)an+1 + pqan ≥ 0 , ( ) n ≥ 0.

Let K be set of all real sequences, K+ the set of all real sequences positive, Kp, q

the set of all p, q-convex sequences and T : Kp, q → K be a linear operator, defined as:

T(a; n) =

∑

= ⋅ +

n

k

k s k n a 0

) (

ρ

where ρk(n) R (n N) and s N are arbitrary.

The purpose of this work is to determine sufficient conditions for a real triangular matrix ‖ρk(n)‖n N, k=0, …, nsuch that T(Kp, q) K+.

Theorem 1

Let a = (an)n≥0 Kp, q be given arbitrary. T(a; n) K+ if

i)

      

= ⋅

= ⋅

∑

∑

= =

0 )

(

0 ) (

0 0 n

i

i i n

i

i i

p n

q n

ρ ρ

p≠q , p≠0 , q≠0

or

      

= ⋅ ⋅

= ⋅

∑

∑

= =

0 )

(

0 ) (

0 0 n

i

i i

n

i

i i

p i n

q n

ρ ρ

ii)

0 )

( 1

0 0

1  ≥

   ⋅ ⋅

⋅

∑

∑

−

= =

+

i k

r

r k

i

i i

k _q

p p

n p

q ρ , k=0,n−2

Proof. For the computations we need the following notation: ρk(n) = ρk, ( )k =0,n, n N. Consider

T(a; n) =

∑

−

= ⋅ + + − + + + + ⋅ +

2

0

1

2 ( ) ]

[ n

k

k s k

s k

s

k a p q a p qa

c ,

T(a; n) =

∑

∑

∑

−

= + −

= + + −

= ⋅ + + − + ⋅ ⋅ + ⋅ ⋅ ⋅

2

0 2

0

1 2

0

2 ( )

n

k

k s k n

k

k s k n

k

k s

k a p q c a p q c a

c

T(a; n) =

∑

−

= + ⋅ − − + − + ⋅ ⋅

2

0

1

2 ( ) ]

[ n

k

k k

k k

s c p q c p q c

a + as+n-1⋅[c_n-3 - (p+q)c_n-2] +

+ cn-2⋅a_s+n +a_s+1⋅[p⋅qc₁-(p+q)⋅c₀]+p⋅q⋅c₀⋅a_s. Therefore

       

=

= +

−

= +

+ −

= + −

=

−

− − −

− −

n n

n n n

k k k

k

c

c q p c

pqc c

q p c

c q p pqc pqc

ρ

ρ ρ

ρ ρ

2

1 2 3

1 2

1 0 1

0 0

) (

) (

) (

, k=2,n−2 (1)

From (1), it follows:

r r r

r p q c pqc

c ₋₂ −( + ) ₋₁+ =ρ , r=2,n−2 r

r r

r

r pc q c pc

c ₋ − ₋ )− ( ₋ − )=ρ

( _{2 1 1}

2 1

1 1

2

2_{( ) ( )} −

− − −

−

− _{− − − =} r

r r r

r r

r

r _{c pc q c pc q}

q ρ

c0 – pc-qk-1⋅(ck-1 - p⋅ck) =

∑

= − ⋅ k r r r q 2 2

ρ , k=2,n−2

p⋅ck - ck-1 = 1 0 1 − − k q c pc

+ _k1₋₁

q

∑

₌

− ⋅ k r r r q 2 2 ρ ,

pk⋅ck - p

k-1

⋅ck-1 = (pc₁-c₀)⋅

1 −     k q p + 1 −     k q p

∑

₌k _⋅ − r r r q 2 2 ρ ,

pi⋅c_i – p

i-1

⋅c_i-1 = (pc₁-c₀)⋅

1 −     i q p + 1 −     i q p

∑

₌i _⋅ −

r r r q 2 2 ρ .

Let add those equalities for i = 2, …, k. We obtain:

pk⋅c_k - pc₁ = (pc₁-c₀)

∑

= −     k i i q p 2 1 + ⋅    

∑

₌k −

i i q p 2 1

∑

₌i _⋅ −

r r r q 2 2 ρ

ck = ₋₁ +

1 k q c k p c

pc₁− ₀

∑

= −









k i i

q

p

2 1 + 1 1 − k

qp  ⋅

  

∑

₌k

i i

q p

2

∑

=

⋅ i r r r q 2 ρ

By the formula

∑

⋅

=

k

i i

a

2

∑

= i r r b 2 =

∑

⋅ = k i i b

2

∑

= k

i r

r

a we can write:

ck = ₋₁ +

1 k p c k p c

pc₁− ₀

∑

= −









k i i

q

p

2 1 + 1 1 − k

qp

∑

₌ ⋅

k i i i q 2 ρ

∑

=     k i r r q p

ck = ₋₁ +

1 k p c k p c

pc₁− ₀

∑

= −









k i i

q

p

2 1 + 1 1 − k

qp

∑

₌ ⋅

k i i i q 2 ρ

∑

− = +     i k r i r q p 0

, k=2,n−2

c0 = q p⋅

0

ρ _{, c} 1 =

q p⋅ 1 ρ ₊ 0 2 2_⋅ ⋅ρ

+ q p q p Therefore

ck = +

⋅_pk

q

1

ρ

0 1 2 ⋅ρ

+ + k p q q p + k p 1         −     ⋅ ⋅ + + pq q p q p pq

p ₀ 0

2 2

1 ρ ρ

ρ

∑

= −     k i i q p 2 1 + + 1 1 + k

qp

∑

₌ ⋅

k i i i q 2 ρ

∑

− = +     i k r i r q p 0 ,

ck = 1 1

+

k

qp

∑

₌ ⋅ ⋅

k i i i q 0 ρ

∑

− = +     i k r i r q p 0

, k=2,n−2 (1’)

From (1) and (1’) it follows that:

       + + =     ⋅ = =     ⋅ =

∑

∑

∑

∑

− − = − + − = − − − − = + − = − − n i n r n i r n i i i n n i n r n i r n i i i n n q p q p q qp c q p q qp c ρ ρ ρ ρ ρ ) ( 1 1 3 0 1 3 0 2 3 2 0 2 0 1 2 (1’’) Because n i n n i i i n

n q w

qp

c = ρ ⋅ ⋅ ₋₋ =ρ

−

= −

−

∑

1

2 0 1 2 1 where

( )

      = ≠ − − = q p j q p w q p j q p j , , 1 1

it results:

n q

p p q

n n n i n n

i

i i n

n

qp p w

p qp

c ρ ρ =ρ

− − ⋅ −

⋅ ⋅

= −− ₋

= −

−

∑

1 1 1

1 1

0 1

2 .

Thus:

0 1 0

= ⋅

⋅ −− =

∑

n i

n

i

i i p w

ρ (2)

Because

n n

i n i n

i i n

n p w p q

qp

c 1 ρ ₂ ρ ₁ ( )ρ

3

0 2

3= ⋅ − − = − + +

−

= −

−

∑

it results also

1 1 1

1

1 2

2

2 1

2 0

2 3

− − ⋅ −

− − ⋅ −

⋅

= −

− − =

−

−

∑

q p p q

n

q p p q n i n n

i i i n

n

q p q

p w

p qp

c ρ ρ ρ

and thus:

0 2 0

= ⋅

⋅ − −

=

∑

n i

n

i

i i p w

ρ (3)

From (2) and (3) we obtain:

      

= ⋅

= ⋅

∑

∑

= =

0 0

0 0 n

i

i i n

i

i i

p q

ρ ρ

p≠q

or

      

= ⋅ ⋅

= ⋅

∑

∑

= =

0 0

0 0 n

i

i i n

i

i i

p i q

ρ ρ

this means i).

From condition ck ≥ 0 for all k from 0 to n-2 we obtain ii).

References

[1] Hard, G. H., Divergent Series, Oxford, 1949. [2] Popoviciu T., Let functions convexes, Paris, 1944.

[3] I. Z. Milovanović, J. E. Pecárić, Gh. Toader, On p,q-convex sequences, Itinerant seminar on functional equations, approximation and convexity, Cluj Napoca, 1985.

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