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volume 7, issue 5, article 193, 2006.

Received 29 June, 2006;

accepted 09 November, 2006.

Communicated by:K. Nikodem

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Journal of Inequalities in Pure and Applied Mathematics

A GENERALIZATION OF HÖLDER AND MINKOWSKI INEQUALITIES

YILMAZ YILMAZ, M. KEMAL ÖZDEM˙IR AND ˙IHSAN SOLAK

Department of Mathmematics

˙Inönü University

44280 Malatya / TURKEY EMail:yyilmaz@inonu.edu.tr EMail:kozdemir@inonu.edu.tr EMail:isolak@inonu.edu.tr

c

2000Victoria University ISSN (electronic): 1443-5756 177-06

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A Generalization of Hölder and Minkowski Inequalities

Yılmaz Yilmaz, M. Kemal Özdemir and ˙Ihsan Solak

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J. Ineq. Pure and Appl. Math. 7(5) Art. 193, 2006

http://jipam.vu.edu.au

Abstract

In this work, we give a generalization of Hölder and Minkowski inequalities to normal sequence algebras with absolutely monotone seminorm. Our main re- sult is Theorem2.1 and Theorem2.2 which state these extensions. Taking F =`₁andk·k_F =k·k₁in both these theorems, we obtain classical versions of these inequalities. Also, using these generalizations we construct the vector- valued sequence spaceF(X, λ, p)as a paranormed space which is a most general form of the spacec₀(X, λ, p)investigated in [6].

2000 Mathematics Subject Classification: Primary 26D15, 47A30; Secondary 46A45.

Key words: Inequalities, Hölder, Minkowski, Sequence algebra, Vector-valued se- quence space.

1. Introduction

Hölder and Minkowski inequalities have been used in several areas of mathematics, especially in functional analysis. These inequalities have been generalized in various directions. The purpose of this paper is to give some extensions of the classical Hölder and Minkowski inequalities. We discovered that the classical versions are only a type of these extensions in `₁ which is a normal sequence algebra with absolutely monotone seminormk·k₁.

We now recall some definitions and facts.

A Frechet space is a complete total paranormed space. IfH is an Hausdorff space then an FH-space is a vector subspaceX ofH which is a Frechet space and is continuously embedded inH, that is, the topology ofXis larger than the relative topology ofH. Moreover if X is a normed FH-space then it is called a BH-space. An FH-space with H = w, the space of all complex sequences, is called an FK-space, so a BK-space is a normed FK-space. We know that

`∞, c, c₀ and `_p (1 ≤ p < ∞) are BK-spaces. The following relation exists among these sequence spaces:

`_p ⊂c₀ ⊂c⊂`∞.

A basis for a topological vector spaceX is a sequence(bn)such that every x ∈ X has a unique representation x = P

t_nb_n. This is equivalent to the fact thatx−Pm

n=1t_nb_n →0 (m → ∞)in the vector topology ofX. For example, c0 and`p have (en) as a basis (en is a sequence xwhere xn = 1, xk = 0for n 6= k). If X has a basis (b_n) the functionals l_n, given by l_n(x) = t_n when x = P

t_nb_n, are linear. They are called the coordinate functionals and(b_n)is called a Schauder basis if each ln ∈ X⁰, the continuous dual ofX. A basis of

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a Frechet space must be a Schauder basis [7]. An FK-spaceX is said to have AK, or be an AK-space, if X ⊃ φ (the space of all finite sequences) and(en) is a basis for X, i.e. for eachx, x^[n] → x, wherex^[n], the nth section of xis Pn

k=1x_ke_k; otherwise expressed,x=P

x_ke_kfor allx∈X [8]. The spacesc₀ and`p are AK-spaces butcand`∞ are not. We say that a sequence spaceF is an AK-BK space if it is both a BK and an AK-space.

An algebraA over a fieldK is a vector space A overK such that for each ordered pair of elementsx, y ∈ Aa unique productxy∈ Ais defined with the properties

(1)(xy)z =x(yz) (2a)x(y+z) =xy+xz (2b)(x+y)z =xz+yz (3)α(xy) = (αx)y=x(αy) for allx, y, z ∈Aand scalarsα[4].

If K = R (real field) or C (complex field) then A is said to be a real or complex algebra, respectively.

LetF be a sequence space andx, y be arbitrary members ofF. F is called a sequence algebra if it is closed under the multiplication defined by xy = (x_iy_i), i ≥ 1, and is called normal or solid ify ∈ F whenever |y_i| ≤ |x_i|, for some x ∈ F. If F is both a normal and sequence algebra then it is called a normal sequence algebra. For example,cis a sequence algebra but not normal.

w, `_∞, c₀and`_p (0< p <∞)are normal sequence algebras.

A paranormpon a normal sequence spaceF is said to be absolutely monotone ifp(x)≤p(y)forx, y ∈F with|x_i| ≤ |y_i|for eachi[3].

The normkxk_∞ = sup|x_k|which makes the spaces`∞, c, c₀a BK-space, is absolutely monotone. For p ≥ 1, the norm kxk = (P∞

k=1|xk|^p)^1/p over `p is

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absolutely monotone. Also, for0< p <1, thep-normkxk_p =P∞

k=1|x_k|^pover

`pis absolutely monotone.

An Orlicz function is a functionM : [0,∞)−→[0,∞)which is continuous, non-decreasing and convex withM(0) = 0,M(x)>0forx >0andM(x)→

∞ as x → ∞. We say that the Orlicz function M satisfies the ∆´-condition if there exist positive constants a and u such that M(xy) ≤ aM(x)M(y) (x, y ≥u). By means of M, Lindenstrauss and Tzafriri [2] constructed the sequence space

`_M =

x∈w:X M

|x_k| ρ

<∞for someρ >0

with the norm kxk_M = infn

ρ >0 :P M_|x

k| ρ

≤1o

. This norm is absolutely monotone and`_M is normal sinceM is non-decreasing. Also ifM satisfies the∆´-condition then`_M is a sequence algebra.

Now we give a useful inequality from classical analysis.

Lemma 1.1. Let f be a function such that f⁰⁰(x) ≥ 0 for x > 0. Then for 0< a < x < b

f(x)−f(a)

x−a = 1

x−a Z x

a

f⁰(t)dt

≤f⁰(x)

≤ 1 b−x

Z b x

f⁰(t)dt

= f(b)−f(x) b−x .

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Hence

f(x)≤ b−x

b−af(a) + x−a b−af(b) [7].

Apply this to the functionf(x) =−lnxwithθ = (b−x)/(b−a). Then for alla, bpositive numbers and0≤θ≤1, we have

(1.1) a^θb^1−θ ≤aθ+ (1−θ)b.

Next, we give a lemma associated with the theorems in Section2.

Lemma 1.2.

a) Let F be a normal sequence algebra, u = (un) ∈ F andp ≥ 1. Then u^p = (u^p_n)∈F.

b) IfF is a normal sequence space,k·k_F is an absolutely monotone seminorm onF andu= (un)∈F then|u|= (|un|)∈F andk|u|k_F =kuk_F. Proof. a) We define two sequencesa= (an)andb = (bn)such that

a_n =

( u_n if |u_n| ≥1 0 if |u_n|<1

and b_n=

( 0 if |u_n| ≥1 u_n if |u_n|<1

.

Sou_n=a_n+b_nandu^p_n =a^p_n+b^p_n. Obviously,a, b∈F. Sincep < [p] + 1, we have

|a_n|^p ≤ |a_n|^[p]+1,

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where [p] denotes the integer part of p. Since F is a sequence algebra, the sequence a^[p]+1 is a member ofF by induction, and so a^p ∈ F. Furthermore, sinceF is normal and|b_n|^p ≤ |b_n|, we haveb^p ∈F. Henceu^p ∈F.

b) It is a direct consequence of normality and absolute monotonicity.

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2. Generalizations

Our main results are the following theorems which state the extensions of Hölder and Minkowski inequalities. TakingF =`₁ andk·k_F =k·k₁ in both Theorem 2.1 and Theorem 2.2, we get classical versions of these inequalities. More- over, if we change the choices ofF andk·k_F then we can obtain many different inequalities corresponding to these generalizations. Therefore, the following results are quite productive.

Theorem 2.1. LetF be a sequence algebra andk·k_F be an absolutely monotone seminorm onF. Supposeu= (u_n), v= (v_n)∈F. Then

kuvk_F ≤ ku^pk^1/p_F kv^qk^1/p_F , wherep >1and ¹_p + ¹_q = 1.

Proof. Assume that x_n = |u_n|^p andy_n = |v_n|^q. It is immediate from Lemma 1.2(a) that x = (x_n) and y = (y_n) are members of F. Let M = kxk_F and N =kyk_F. Then it follows from inequality (1.1) that for eachn,

x_n M

θy_n N

1−θ

≤θx_n

M + (1−θ)y_n N

as0≤θ≤1. Becausek·k_F is an absolutely monotone seminorm we write

x_n M

θy_n N

1−θ F

≤

θx_n

M + (1−θ)y_n N

F

. Hence

1 M^θN^1−θ

x^θ_ny_n^1−θ F ≤1,

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so that

x^θ_ny^1−θ_n

F ≤ k(xn)k^θ_Fk(yn)k^1−θ_F . Settingθ= 1/p, we get

x^1/p_n y^1/q_n

F ≤ k(x_n)k^1/p_F k(y_n)k^1/q_F , and puttingxn=|un|^pandyn =|vn|^q, we obtain

k(|unvn|)k_F ≤ k(|un|^p)k^1/p

F k(|vn|^q)k^1/q

F . So, it follows from Lemma1.2(b) that

kuvk_F ≤ ku^pk^1/p

F kv^qk^1/q

F .

Theorem 2.2. LetF be a normal sequence algebra andk·k_F be an absolutely monotone seminorm onF. Then for everyu= (u_n), v = (v_n)∈F andp≥1,

k(u+v)^pk^1/p_F ≤ ku^pk^1/p_F +kv^pk^1/p_F , where(u+v)^p = ((u_n+v_n)^p).

Proof. Forp= 1, it is obvious.

Letp >1. Proceeding with the manner of the proof in the classical version, we write

(u+v)^p =u(u+v)^p−1+v(u+v)^p−1.

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It follows from Theorem2.1that k(u+v)^pk_F ≤ ku^pk^1/p_F

(u+v)^(p−1)q

1/q F

+kv^pk^1/p_F

(u+v)^(p−1)q

1/q

F

=

ku^pk^1/p_F +kv^pk^1/p_F

(u+v)^(p−1)q

1/q F , where¹_p+¹_q = 1. Hence, dividing the first and last terms by

(u+v)^(p−1)q

1/q

F =

k(u+v)^pk^1/q_F , we obtain the inequality.

Example 2.1. Taking F = `∞ and k·k_F = k·k_∞ in both Theorem 2.1 and Theorem2.2, we obtain the inequalities

sup

n

|u_nv_n| ≤

sup

n

|u_n|^p ¹_p

·

sup

n

|v_n|^q ¹_q

and

sup

n

|u_n+v_n|^p ¹_p

≤

sup

n

|u_n|^p ¹_p

+

sup

n

|v_n|^p ¹_p

,

where u, v ∈ `∞ and ¹_p + ¹_q = 1as p > 1. Hence, in fact, these elementary inequalities are extended Hölder and Minkowski inequalities respectively.

Example 2.2. Now putF =`M andk·k_F =k·k_M in Theorem2.1and Theorem 2.2, whereM satisfies the∆´-condition. In this case, we write the inequalities

inf

ρ >0 :X M

|x_ky_k| ρ

≤1

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≤

inf

ρ >0 :X M

|x_k|^p ρ

≤1 ¹_p

·

inf

ρ >0 :X M

|y_k|^q ρ

≤1 ¹_q

and

inf

ρ >0 :X M

|x_k+y_k|^p ρ

≤1 ¹_p

≤

inf

ρ >0 :X M

|x_k|^p ρ

≤1 ¹_p

+

inf

ρ >0 :X M

|y_k|^p ρ

≤1 ¹_p

as Hölder and Minkowski inequalities respectively.

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3. An Application

Now let us introduce the class F(X, λ, p) of vector-valued sequence spaces which includes the space c0(X, λ, p)investigated in [6] with some linear topological properties. Theorem2.2makes it possible to improve some topological properties of the spaceF (X, λ, p).

Let F be an AK-BK normal sequence algebra such that the norm k·k_F of F is absolutely monotone and X be a seminormed space. Also suppose that λ = (λ_k)is a non-zero complex sequence andp= (p_k)is a sequence of strictly positive real numbers. Define the vector-valued sequence class

F(X, λ, p) ={x∈s(X) : ([q(λ_kx_k)]^p^k)∈F},

whereqis the seminorm ofXands(X)is the most generalX-termed sequence space. F (X, λ, p)becomes a linear space under natural co-ordinatewise vector operations if and only if p∈`∞(see Lascarides [1]). TakingF = c₀ andXas a Banach space we get the spacec₀(X, λ, p)in [6].

Lemma 3.1. Let0< t_k ≤1. Ifa_kandb_kare complex numbers then we have

|a_k+b_k|^t^k ≤ |a_k|^t^k +|b_k|^t^k [5, p.5].

Lemma 3.2. Let(X, q)be a seminormed space, andF a normal AK-BK space with an absolutely monotone norm k·k_F. Supposep = (p_k)is a bounded se- quence of positive real numbers. Then the map

ex_n: [0,∞)→[0,∞) ; ex_n(u) =

n

X

k=1

[uq(λ_kx_k)]^p^ke_k F

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defined by means ofx = (x_k)∈F (X, λ, p)and a positive integern, is contin- uous, where(ek)is a unit vector basis ofF.

Proof. Since the norm function is continuous it is sufficient to show that the mappings defined by

g_k : [0,∞)→F, g_k(u) = [uq_k(λ_kx_k)]^p^ke_k are continuous. Letui →0 (i→ ∞), then

g_k(u_i)→(0,0, . . .) (i→ ∞)

for eachk. Hence, eachg_kis sequential continuous (it is equivalent to continuity here).

Theorem 3.3. Define the functiong :F (X, λ, p)−→Rby g(x) = k([q(λ_kx_k)]^p^k)k^1/M_F ,

whereM = max (1,suppn). Thengis a paranorm onF (X, λ, p).

Proof. It is obvious thatg(θ) = 0andg(−x) =g(x). From the absolute monotonicity ofk·k_F, Lemma3.1and Theorem2.2, we get

g(x+y) =

[q(λ_kx_k+λ_ky_k)]^p^k^/MM

1/M

F

≤

[q(λ_kx_k)]^p^k^/M + [q(λ_ky_k)]^p^k^/MM

1/M

F

≤ k([q(λ_kx_k)]^p^k)k^1/M_F +k([q(λ_ky_k)]^p^k)k^1/M_F

=g(x) +g(y)

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forx, y ∈F (X, λ, p).

To show the continuity of scalar multiplication assume that (µⁿ) is a sequence of scalars such that|µⁿ−µ| → 0 (n→ ∞)andg(xⁿ−x)→0 (n → ∞) for an arbitrary sequence(xⁿ)⊂F(X, λ, p). We shall show that

g(µⁿxⁿ−µx)→0 (n→ ∞). Sayτ_n =|µⁿ−µ|and we get

g(µⁿxⁿ−µx) = k([q(λ_k(µⁿxⁿ_k −µx_k))]^p^k)k^1/M_F

=k([q(λ_k(µⁿxⁿ_k −µⁿx_k+µⁿx_k−µx_k))]^p^k)k^1/M_F

≤ k([|µⁿ|q(λk(xⁿ_k−xk))) +τnq(λkxk)]^p^k)k^1/M_F

≤

n

[A(k, n)]^p^k^/M + [B(k, n)]^p^k^/MoM

1/M

F

, where A(k, n) = Rq(λ_k(xⁿ_k−x_k)), B(k, n) = τ_nq(λ_kx_k) and R = max{1,sup|µⁿ|}. Again by Theorem2.2we can write

g(µⁿxⁿ−µx)≤ k(A(k, n))k^1/M_F +k(B(k, n))k^1/M_F

≤R

A R

p_k

1/M

F

+k(B(k, n))k^1/M_F

=Rg(xⁿ−x) +k(B(k, n))k^1/M_F .

Since g(xⁿ−x) → 0 (n→ ∞) we must show that k(B(k, n))k^1/M_F → 0 (n → ∞). We can find a positive integern0 such that0≤ τn ≤ 1forn ≥n0.

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Sayt_k = [q(λ_kx_k)]^p^k. Sincet = (t_k)∈F andF is an AK-space, we get

t−

m

X

k=1

t_ke_k F

=

∞

X

k=m+1

[q(λ_kx_k)]^p^ke_k F

→0 (m → ∞),

where(e_k)is a unit vector basis ofF. Therefore, for everyε >0there exists a positive integerm₀ such that

∞

X

k=m0+1

[q(λkxk)]^p^kek

1 M

F

< ε 2.

Forn ≥ n₀ write[(τ_nq(λ_kx_k))]^p^k ≤ [f(q(λ_kx_k))]^p^k for eachk. On the other hand, we can write

∞

X

k=m0+1

[τ_nq(λ_kx_k)]^p^ke_k

1 M

F

≤

∞

X

k=m0+1

[q(λ_kx_k)]^p^ke_k

1 M

F

< ε 2. Now, from Lemma3.2, the function

ex_m₀(u) =

m0

X

k=1

[(uq(λ_kx_k))]^p^ke_k F

is continuous. Hence, there exists aδ(0< δ <1)such that ex_m₀(u)≤ε

2 M

,

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for0 < u < δ. Also we can find a number∆such thatτ_n < δ forn > ∆. So forn >∆we have

(ex_m₀(τ_n))^1/M =

m0

X

k=1

[τ_nq(λ_kx_k)]^p^ke_k F

< ε 2, and eventually we get

k([τnq(λkxk)]^p^k)k^1/M_F

=

∞

X

k=1

1 M

F

=

m0

X

k=1

[τ_nq(λ_kx_k)]^p^ke_k+

∞

X

k=m0+1

1 M

F

≤

m0

X

k=1

1 M

F

+

∞

X

k=m0+1

1 M

F

< ε 2 + ε

2 =ε.

This shows thatk(B(k, n))k^1/M_F →0 (n→ ∞).

Theorem 3.4. Let(X, q)be a complete seminormed space. ThenF (X, λ, p)is complete with the paranormg. If X is a Banach space thenF (X, λ, p)is an FK-space, in particular, an AK-space.

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Proof. Let(xⁿ)be a Cauchy sequence inF (X, λ, p). Therefore

g(xⁿ−x^m) =k([q(λ_k(xⁿ_k−x^m_k))]^p^k)k^1/M_F →0 (m, n→ ∞), also, sinceF is an FK-space, for eachk

[q(λ_k(xⁿ_k −x^m_k))]^p^k →0 (m, n→ ∞)

and so |λ_k|q(xⁿ_k−x^m_k)→0 (m, n→ ∞). Because of the completeness ofX, there exists anx_k∈ Xsuch thatq(xⁿ_k −x_k)→ 0 (n→ ∞)for eachk. Define the sequence x = (x_k) with these points. Now we can determine a sequence η_k ∈c₀ (0< ηⁿ_k ≤1)such that

(3.1) [|λ_k|q(xⁿ_k −x_k)]^p^k ≤η_kⁿ[q(λ_kxⁿ_k)]^p^k sinceq(xⁿ_k−x_k)→0. On the other hand,

[q(λkxk)]^p^k ≤D{[q(λk(xⁿ_k −xk))]^p^k + [q(λkxⁿ_k)]^p^k}, whereD= max 1,2^H−1

;H = supp_k. From (3.1) we have [q(λ_kx_k)]^p^k ≤D(1 +η_kⁿ) [q(λ_kxⁿ_k)]^p^k

≤2D[q(λ_kxⁿ_k)]^p^k.

So we getx∈F(X, λ, p). Now, for eachε >0there existn₀(ε)such that [g(xⁿ−x^m)]^M < ε^M forn, m > n₀.

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Also, we may write from the AK-property ofF that

m0

X

k=1

[q(λ_k(xⁿ_k−x^m_k))]^p^ke_k F

≤

∞

X

k=1

[q(xⁿ_k −x^m_k)]^p^ke_k F

=

g(xⁿ−x^m)^M . Lettingm→ ∞we have

m0

X

k=1

[q(λ_k(xⁿ_k−x^m_k))]^p^ke_k F

→

m0

X

k=1

[q(λ_k(xⁿ_k−x_k))]^p^ke_k F

< ε^M forn > n₀. Since(e_k)is a Schauder basis forF,

m0

X

k=1

[q(λ_k(xⁿ_k −x_k))]^p^ke_k F

→ k([q(λ_k(xⁿ_k −x_k))]^p^k)k_F

< ε^M asm₀ → ∞.

Then we getg(xⁿ−x)< εforn > n₀sog(xⁿ−x)→0 (n→ ∞).

For the rest of the theorem; we can say immediately that F (X, λ, p) is a Frechet space, becauseXis a Banach space. Also, the projections

Pˆ_k: F (X, λ, p)−→X; Pˆ_k(x) =x_k are continuous sinceP_k=|λ_k|

q◦Pˆ_k

for eachk. WhereP_k’s are coordinate mappings onF and they are continuous sinceF is an FK-space.

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Letx^[n] be thenth section of an element xof F (X, λ, p). We must prove thatx^[n]→xinF (X, λ, p)for eachx∈F (X, λ, p). Indeed,

g x−x^[n]

=g(0,0, . . . ,0, x_n+1, x_n+2, . . .)

=

∞

X

k=n+1

[q(λkxk)]^p^kek

F

→0

sinceF is an AK-space. HenceF (X, λ, p)is an AK-space.

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References

[1] C.G. LASCARIDES, A study of certain sequence spaces of Maddox and a generalization of a theorem of Iyer, Pacific J. Math., 38 (1971), 487–500.

[2] J. LINDENSTRAUSSANDL. TZAFRIRI On Orlicz sequence space, Israel J. Math., 10 (1971), 379–390.

[3] P.K. KAMTHAN AND M. GUPTA, Sequence Spaces and Series, Marcel Dekker Inc., New York and Basel, 1981.

[4] E. KREYSZIG, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1978.

[5] S. NANDAANDB. CHOUDHARY, Functional Analysis with Applications, John Wiley & Sons, New York, 1989.

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