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Stability of the Young and Hölder inequalities

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Abstract

We give a simple proof of the Aldaz stability version of the Young and Hölder inequalities and further refinements of available stability versions of those inequalities.

MSC:26D15.

1 Introduction

In this paper, we study the Young and Hölder inequalities from the point of view of the deviation from equalities with better upper and lower bound estimates. Particularly, we give a further refinement of Aldaz stability type inequalities [1] as well as a simple proof based exclusively on an algebraic argument with the standard Young inequality.

Throughout this paper, the following remainder function [2] plays an important role:

R(θ;a,b)=θa+(1θ)b a θ b 1 θ ,
(1.1)

where a,b>0 and 0θ1.

The standard Young inequality is described as

R(θ;a,b)0,
(1.2)

which may be used without particular comments. The standard Hölder inequality follows from (1.2) and the equality

Ω |fg|dμ= f p g p ( 1 Ω R ( 1 p ; | f | p f p p , | g | p g p p ) d μ )
(1.3)

for all f L p (Ω,μ){0} and g L p (Ω,μ){0}, where L q (Ω,μ) is the Banach space of q th integrable functions on a measure space (Ω,μ) with the norm q , 1<q<, and p is the dual exponent of p defined by 1/p+1/ p =1.

The purpose in this paper is to give a clear understanding of the standard Young and Hölder inequalities on the basis of upper and lower bound estimates on the remainder function R(θ;a,b). In Section 2, we reexamine the multiplication formula on R(θ;a,b) [2] and present its dual formula. As a corollary, we give an algebraic proof of Aldaz stability type inequalities for the Young and Hölder inequalities [1]. In Section 3, we compare upper and lower bound estimates on R(θ;a,b) in [14]. In Section 4, we give dyadic refinements of the multiplication formulae on R(θ;a,b) with their straightforward corollaries on (1.3) and discuss the associated dyadic refinements of the Hölder inequality.

There are many papers on the related subjects. We refer the reader to [17] and the references therein.

We close the introduction by giving some notation to be used in this paper. For a,bR we denote by ab and ab their minimum and maximum, respectively.

2 Multiplication formulae

In this section, we revisit the original multiplication formula on R(θ;a,b) [2] in connection with Aldaz stability type inequalities [1]. First of all, we recall Kichenassamy’s multiplication formula.

Proposition 2.1 (Kichenassamy [2])

Let θ and σ satisfy 0θ,σ1. Then the equality

R(θσ;a,b)=θR(σ;a,b)+ b 1 σ R ( θ ; a σ , b σ )
(2.1)

holds for all a,b>0.

Proof The proposition follows from the equality

R ( σ θ , a , b ) = σ θ a + ( 1 θ σ ) b a σ θ b 1 σ θ = θ ( σ a + ( 1 σ ) b a σ b 1 σ ) + θ a σ b 1 σ + ( 1 θ ) b a σ θ b 1 σ θ = θ R ( σ , a , b ) + b 1 σ ( θ a σ + ( 1 θ ) b σ a σ θ b σ ( 1 θ ) ) .

 □

Corollary 2.2 Let θ and σ satisfy 0<θσ<1. Then the equality

R(θ;a,b)= θ σ R(σ;a,b)+ b 1 σ R ( θ σ ; a σ , b σ )
(2.2)

holds for all a,b>0.

Proposition 2.3 Let θ and σ satisfy 0θσ<1. Then the equality

R(σ,a,b)= 1 σ 1 θ R(θ,a,b)+ a θ R ( σ θ 1 θ , a 1 θ , b 1 θ )
(2.3)

holds for all a,b>0.

Remark 2.1 Equality (2.3) is regarded as a dual formula for R(θ;a,b) in the sense that 1 σ 1 θ + σ θ 1 θ =1.

Proof of Proposition 2.3

R ( σ , a , b ) = σ a + ( 1 σ ) b a σ b 1 σ = 1 σ 1 θ ( θ a + ( 1 θ ) b a θ b 1 θ ) + σ θ 1 θ a + 1 σ 1 θ a θ b 1 θ a σ b 1 σ = 1 σ 1 θ R ( θ , a , b ) + a θ ( σ θ 1 θ a 1 θ + 1 σ 1 θ b 1 θ a σ θ b ( 1 θ ) ( 1 σ θ 1 θ ) ) .

 □

Corollary 2.4 Let θ and σ satisfy 0θσ<1. Then the equality

R(θ;a,b)= 1 θ 1 σ R(σ;a,b) 1 θ 1 σ a θ R ( σ θ 1 θ ; a 1 θ , b 1 θ )
(2.4)

holds for all a,b>0.

Remark 2.2 Propositions 2.1 and 2.3 are equivalent. In fact, it follows from the reciprocal formula R(θ;a,b)=R(1θ,b,a) and Proposition 2.1 that

R ( σ ; a , b ) = R ( 1 σ ; b , a ) = 1 σ 1 θ R ( 1 θ ; b , a ) + a 1 ( 1 θ ) R ( 1 σ 1 θ ; b 1 θ , a 1 θ ) = 1 σ 1 θ R ( θ ; a , b ) + a θ R ( σ θ 1 θ ; a 1 θ , b 1 θ ) ,

which is precisely (2.3). Conversely, given θ and σ with 0<θ1, 0<σ1, we put θ =1θσ and σ =1σ. Then we have 0 σ θ <1, σ=1 σ , θ=(1 θ )/(1 σ ), and θσ=1 θ . By the reciprocal formula and Proposition 2.3, we have

R ( θ σ ; a , b ) = R ( 1 θ ; a , b ) = R ( θ ; b , a ) = 1 θ 1 σ R ( σ ; b , a ) + b σ R ( θ σ 1 σ ; b 1 σ , a 1 σ ) = 1 θ 1 σ R ( 1 σ ; a , b ) + b σ R ( 1 θ 1 σ ; a 1 σ , b 1 σ ) = θ R ( σ ; a , b ) + b 1 σ R ( θ ; a σ , b σ ) ,

which is precisely (2.1).

Proposition 2.5 (Aldaz [1], Kichenassamy [2])

Let 0θ1. Then the inequalities

( θ ( 1 θ ) ) ( a 1 / 2 b 1 / 2 ) 2 R(θ;a,b) ( θ ( 1 θ ) ) ( a 1 / 2 b 1 / 2 ) 2
(2.5)

hold for all a,b>0.

Proof Though the first inequality of (2.5) is shown in [2], we show the inequalities in (2.5) for completeness. In the case 0θ1/2, we use Corollaries 2.2 and 2.4 with σ=1/2 to obtain

θ ( a 1 / 2 b 1 / 2 ) 2 = 2 θ R ( 1 / 2 ; a , b ) = R ( θ ; a , b ) b 1 / 2 R ( 2 θ ; a , b ) R ( θ ; a , b ) = 2 ( 1 θ ) R ( 1 / 2 ; a , b ) 2 ( 1 θ ) a θ R ( 1 / 2 θ 1 θ ; a 1 θ , b 1 θ ) 2 ( 1 θ ) R ( 1 / 2 ; a , b ) = ( 1 θ ) ( a 1 / 2 b 1 / 2 ) 2 .
(2.6)

In the case 1/2θ1, we apply (2.6) with θ replaced by 1θ to obtain

2(1θ)R(1/2;b,a)R(1θ;b,a)2θR(1/2;b,a),

which is precisely (2.5). □

Remark 2.3 An equivalent couple of inequalities in Proposition 2.5 were proved by Aldaz [1] by differential calculus. The proof above depends on algebraic identities with the standard Young inequality.

3 Upper and lower bounds of the remainder function

In this section, we collect and compare several bounds of the remainder function R(θ;a,b). For that purpose, we study the upper and lower bound estimates in terms of majorant M(θ;a,b) and minorant m(θ;a,b) in the form

m(θ;a,b)R(θ;a,b)M(θ;a,b)

for all a,b>0. We introduce four couples of the bounds as follows:

[ A ] m A ( θ ; a , b ) = ( θ ( 1 θ ) ) ( a 1 / 2 b 1 / 2 ) 2 , M A ( θ ; a , b ) = ( θ ( 1 θ ) ) ( a 1 / 2 b 1 / 2 ) 2 , [ K ] m K ( θ ; a , b ) = θ ( 1 θ ) 2 ( a b ) ( log a log b ) 2 , M K ( θ ; a , b ) = θ ( 1 θ ) 2 ( a b ) ( log a log b ) 2 , [ H ] m H ( θ ; a , b ) = ( θ ( 1 θ ) ) | a θ ( 1 θ ) b θ ( 1 θ ) | 1 / ( θ ( 1 θ ) ) , M H ( θ ; a , b ) = ( θ ( 1 θ ) ) | a θ ( 1 θ ) b θ ( 1 θ ) | 1 / ( θ ( 1 θ ) ) , [ FO ] m F O ( θ ; a , b ) = θ ( 1 θ ) 2 ( a b ) ( a b ) 2 , M F O ( θ ; a , b ) = θ ( 1 θ ) 2 ( a b ) ( a b ) 2 .

Those couples are given respectively in [1, 2, 4], and [3].

Remark 3.1 By the monotonicity property suggested in [2], the remainder function with respect to θ[0,1] is approximated arbitrarily precisely by the remainder functions with respect to rationals which approximate θ. However, the approximation obtained by the monotonicity property is rather involved. Here, we focus only on lower and upper bounds with regard to a difference.

Simple relationships in those couples are summarized in the following.

Proposition 3.1 Let 0θ1. Then the inequalities

m H (θ;a,b) m A (θ;a,b)R(θ,a,b) M A (θ;a,b) M H (θ,a,b),
(3.1)
m K (θ;a,b) m F O (θ;a,b)R(θ,a,b) M K (θ;a,b) M F O (θ,a,b)
(3.2)

hold for all a,b>0.

Proof By homogeneity, (3.1) follows from the inequality

( x θ 1 ) 1 / θ ( x σ 1 ) 1 / σ
(3.3)

for all x1 and any θ and σ with 0θσ. Inequality (3.3) follows from

x θ = ( x σ 1 + 1 ) θ / σ ( x σ 1 ) θ / σ +1.

Although some inequalities in (3.2) are proved in [4, 8], we prove (3.2) for completeness. By the integral representations [4, 8]

R ( θ ; a , b ) = θ ( 1 θ ) [ 0 1 0 t ( t a + ( 1 t ) b ) θ 1 ( s a + ( 1 s ) b ) θ d s d t ] ( a b ) 2 = [ 0 1 ( ( θ ( 1 t ) ) ( ( 1 θ ) t ) ) a t b 1 t d t ] ( log a log b ) 2 ,

we have

m F O ( θ ; a , b ) R ( θ ; a , b ) M F O ( θ ; a , b ) , m K ( θ ; a , b ) R ( θ ; a , b ) M K ( θ ; a , b ) .

Then it suffices to prove that

m K ( θ ; a , b ) m F O ( θ ; a , b ) , M K ( θ ; a , b ) M F O ( θ ; a , b ) .

The last two inequalities are equivalent and follow from

x ( log x ) 2 ( x 1 ) 2

for all x>0. □

Proposition 3.2 Let 0<θ<1 and let

t 0 (θ)= ( 2 ( θ ( 1 θ ) ) 1 ) 2 .

Then the following inequalities hold for all a,b>0:

m A (θ,a,b) m F O (θ,a,b)if (ab) t 0 (θ)ab,
(3.4)
m A (θ,a,b) m F O (θ,a,b)if 0<ab(ab) t 0 (θ).
(3.5)

Remark 3.2 Since 0<θ(1θ)1/2θ(1θ)<1, t 0 (θ) satisfies

( 2 1 ) 2 < t 0 (θ)1

for all θ with 0θ1. Proposition 3.2 shows that m F O (θ;a,b) is better than m A (θ;a,b) in a neighborhood of the diagonal a=b in the quarter plane (0,)×(0,).

Proof of Proposition 3.2 It is sufficient to show inequalities (3.4) and (3.5) with 0<a<b. We have

lim a 0 m A ( θ , a , b ) = ( θ ( 1 θ ) ) b lim a 0 m F O ( θ , a , b ) = θ ( 1 θ ) 2 b , lim a b m A ( θ , a , b ) ( a 1 / 2 b 1 / 2 ) 2 = θ ( 1 θ ) lim a b m F O ( θ , a , b ) ( a 1 / 2 b 1 / 2 ) 2 = 2 θ ( 1 θ ) .

Moreover, m A (θ,a,b)= m F O (θ,a,b) is equivalent to the equation

( ( a / b ) 1 / 2 + 1 ) 2 = 2 ( θ ( 1 θ ) ) θ ( 1 θ ) .
(3.6)

Since the ratio of a/b satisfying (3.6) with given θ is uniquely determined, inequalities (3.4) and (3.5) follow. □

To compare M A and M K , we prepare Lambert’s W function, which is defined as the inverse function of [1,)xx e 1 / x [1/e,). For details, see [8].

Proposition 3.3 Let 0θ1 and let

t 1 (θ)= 2 ( θ ( 1 θ ) ) W ( 1 2 ( θ ( 1 θ ) ) exp ( 1 2 ( θ ( 1 θ ) ) ) ) ,

where t 1 (0) and t 1 (1) are understood to be

lim θ 0 t 1 (θ)= lim θ 1 t 1 (θ)=1.

Then the following inequalities hold for any a,b>0:

M A (θ;a,b) M K (θ;a,b)if (ab) t 1 (θ)(ab),
(3.7)
M A (θ;a,b) M K (θ;a,b)if (ab) t 1 (θ)(ab).
(3.8)

Remark 3.3 Since 0θ(1θ)1/2, t 1 (θ) satisfies 0 t 1 (θ)1 for 0θ1. In the proof below, we see that 0< t 1 (θ)<1 if 0<θ<1. Proposition 3.3 shows that M K (θ;a,b) is better than M A (θ) in a neighborhood of the diagonal a=b in the quarter plane (0,)×(0,).

Proof of Proposition 3.3 Let t>0 satisfy t 2 =(ab)/(ab). The magnitude correlation of M A (θ,a,b) and M K (θ;a,b) coincides with that of

( a b ) 1 M A ( θ , a , b ) =1t

and

( a b ) 1 M A ( θ , a , b ) = 2 ( θ ( 1 θ ) ) logt.

Let f(t)=1t+ 2 ( θ ( 1 θ ) ) log(t). We have f( t 1 (θ))=0 since

t 1 ( θ ) 2 ( θ ( 1 θ ) ) exp ( t 1 ( θ ) 2 ( θ ( 1 θ ) ) ) = 1 2 ( θ ( 1 θ ) ) exp ( 1 2 ( θ ( 1 θ ) ) ) ,

which is rewritten as

exp ( 1 t 1 ( θ ) 2 ( θ ( 1 θ ) ) ) = t 1 ( θ ) 1 ,

and, moreover,

1 t 1 (θ)= 2 ( θ ( 1 θ ) ) log ( t 1 ( θ ) ) .

In addition,

f (t)=1+ 2 ( θ ( 1 θ ) ) /t.

Then inequalities (3.7) and (3.8) follow from the Table 1.  □

Table 1 The signs at the important values of x

4 Dyadic refinements of multiplication formulae and their applications

In this section, we give dyadic refinements of the multiplication and dual multiplication formulae on the remainder function R(θ;a,b) and their applications. By the reciprocal formula R(θ;a,b)=R(1θ;b,a), it is important to describe the formation of the remainder function as θ0 and θ1/2 with the principal terms 2θR(1/2;a,b) and 2(1θ)R(1/2;a,b). For that purpose, we utilize dyadic decomposition.

Proposition 4.1 Let θ satisfy 0<θ 2 n with an integer n1. Then the equality

R(θ,a,b)=θ j = 1 n 2 j 1 b 1 2 1 j ( a 2 j b 2 j ) 2 + b 1 2 n R ( 2 n θ , a 2 n , b 2 n )
(4.1)

holds for all a,b>0.

Proof We apply Corollary 2.2 with σ=1/2 to obtain

R ( θ ; a , b ) = θ ( a 1 / 2 b 1 / 2 ) + b 1 / 2 R ( 2 θ ; a 1 / 2 , b 1 / 2 ) , b 1 2 j R ( 2 j θ ; a 2 j , b 2 j ) = b 1 2 j ( 2 j + 1 θ R ( 1 / 2 ; a 2 j , b 2 j ) + b 2 j 1 R ( 2 j + 1 θ ; a 2 j 1 , b 2 j 1 ) ) = 2 j θ b 1 2 j ( a 2 j b 2 j ) 2 + b 1 2 j 1 R ( 2 j + 1 θ ; a 2 j 1 , b 2 j 1 )

for any j with 1jn. Then (4.1) follows immediately. □

Proposition 4.2 Let θ satisfy ( 2 m 1 1)/( 2 m 1)θ1/2 with an integer m1. Then the equality

R ( θ ; a , b ) = ( 1 θ ) ( a 1 / 2 b 1 / 2 ) 2 ( 1 2 θ ) j = 1 m 2 j 1 a θ b ( 1 θ ) ( 1 2 1 j ) ( a ( 1 θ ) 2 j b ( 1 θ ) 2 j ) 2 2 ( 1 θ ) a θ b ( 1 θ ) ( 1 2 m ) R ( 2 m 1 / 2 θ 1 θ ; a ( 1 θ ) 2 m , b ( 1 θ ) 2 m )
(4.2)

holds for all a,b>0.

Proof We apply Corollary 2.4 with σ=1/2 to obtain

R(θ;a,b)=(1θ) ( a 1 / 2 b 1 / 2 ) 2 2(12θ) a θ R ( 1 / 2 θ 1 θ ; a 1 θ , b 1 θ ) .
(4.3)

Then (4.2) follows by applying Proposition 4.1 to the last term on the right-hand side of (4.3) with 0(1/2θ)/(1θ) 2 m . □

Corollary 4.3 Let 0θ1/2. Then the inequalities

θ ( a 1 / 2 b 1 / 2 ) 2 + ( 2 θ ( 1 2 θ ) ) b 1 / 2 ( a 1 / 4 b 1 / 4 ) 2 R ( θ ; a , b ) ( 1 θ ) ( a 1 / 2 b 1 / 2 ) 2 ( 1 2 θ ) a θ ( a ( 1 θ ) / 2 b ( 1 θ ) / 2 ) 2 2 ( θ ( 1 2 θ ) ) a θ b ( 1 θ ) / 2 ( a ( 1 θ ) / 4 b ( 1 θ ) / 4 ) 2
(4.4)

hold for all a,b>0.

Corollary 4.4 Let 1/2θ1. Then the inequalities

( 1 θ ) ( a 1 / 2 b 1 / 2 ) 2 + ( ( 2 ( 1 θ ) ) ( 2 θ 1 ) ) a 1 / 2 ( a 1 / 4 b 1 / 4 ) 2 R ( θ ; a , b ) θ ( a 1 / 2 b 1 / 2 ) 2 ( 2 θ 1 ) b 1 θ ( a θ / 2 b θ / 2 ) 2 2 ( ( 1 θ ) ( 2 θ 1 ) ) a ( 1 θ ) / 2 b θ ( a θ / 4 b θ / 4 ) 2
(4.5)

hold for all a,b>0.

Remark 4.1 Some of the lower bounds in Corollaries 4.3 and 4.4 may be found already in [2], Section 3.2.

Remark 4.2 Inequalities (4.4) and (4.5) improve (2.5). Inequalities (2.5) become an equality when θ=1/2, while (4.4) become an equality when θ=0,1/2 and (4.5) become an equality when θ=1/2,1.

We are now in a position to apply the equalities above to Hölder type inequalities.

Theorem 4.5 Let p satisfy 2p< and let m and n be unique integers satisfying

{ 2 n p < 2 n + 1 , n 1 , ( 2 m + 1 1 ) / ( 2 m 1 ) p < ( 2 m 1 ) / ( 2 m 1 1 ) , m 1 .

Then the equalities

f p g p ( 1 1 p Ω ( | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 ) 2 d μ + ( 1 p 1 p ) j = 1 m 2 j 1 Ω | f | f p | g | 1 2 1 j g p 1 2 1 j ( | f | ( p 1 ) 2 j f p ( p 1 ) 2 j | g | 2 j g p 2 j ) 2 d μ + 2 p Ω | f | f p | g | ( 1 2 m ) g p ( 1 2 m ) R ( 2 m 1 p 2 p 1 , | f | ( p 1 ) 2 m f p ( p 1 ) 2 m , | g | 2 m g p 2 m ) d μ ) = f g 1 = f p g p ( 1 1 p j = 1 n 2 j 1 Ω | g | p ( 1 2 1 j ) g p p ( 1 2 1 j ) ( | f | p 2 j f p p 2 j | g | p 2 j g p p 2 j ) 2 d μ Ω | g | p ( 1 2 n ) g p p ( 1 2 n ) R ( 2 n p , | f | p 2 n f p p 2 n , | g | p 2 n g p p 2 n ) d μ )
(4.6)

hold for all f L p (Ω,μ){0} and g L p (Ω,μ){0}.

Proof The theorem follows from (1.3) and Propositions 4.1 and 4.2 with θ=1/p, a= | f | p / f p p , b= | g | p / g p p . □

Corollary 4.6 Let p, m, n be as in Theorem 4.5. Then the inequalities

f p g p ( 1 1 p Ω ( | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 ) 2 d μ + ( 1 p 1 p ) j = 1 m 2 j 1 Ω | f | f p | g | 1 2 1 j g p 1 2 1 j ( | f | ( p 1 ) 2 j f p ( p 1 ) 2 j | g | 2 j g p 2 j ) 2 d μ + 2 p ( 1 2 m 1 p 2 p 1 ) Ω | f | f p | g | ( 1 2 m ) g p ( 1 2 m ) ( | f | ( p 1 ) 2 m 1 f p ( p 1 ) 2 m 1 | g | 2 m 1 g p 2 m 1 ) 2 d μ ) f g 1 f p g p ( 1 1 p j = 1 n 2 j 1 Ω | g | p ( 1 2 1 j ) g p p ( 1 2 1 j ) ( | f | p 2 j f p p 2 j | g | p 2 j g p p 2 j ) 2 d μ p 2 n p Ω | g | p ( 1 2 n ) g p p ( 1 2 n ) ( | f | p 2 n 1 f p p 2 n 1 | g | p 2 n 1 g p p 2 n 1 ) 2 d μ )
(4.7)

hold for all f L p (Ω,μ){0} and g L p (Ω,μ){0}.

Proof The required inequalities follow from Theorem 4.5 and Proposition 2.5. □

Corollary 4.7 Let p 2 n with a positive integer n. Then the inequalities

f p g p ( 1 1 p | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 2 2 + ( 1 p 1 p ) | f | 1 / 2 f p 1 / 2 ( | f | ( p 1 ) / 2 f p ( p 1 ) / 2 | g | 1 / 2 g p 1 / 2 ) 2 2 ) f g 1 f p g p ( 1 1 p j = 1 n 2 j 1 | g | p ( 1 / 2 2 j ) g p p ( 1 / 2 2 j ) ( | f | p 2 j f p p 2 j | g | p 2 j g p p 2 j ) 2 2 p 2 n p | g | p ( 1 / 2 2 n 1 ) g p p ( 1 / 2 2 n 1 ) ( | f | p 2 n 1 f p p 2 n 1 | g | p 2 n 1 g p p 2 n 1 ) 2 2 )
(4.8)

hold for all f L p (Ω,μ){0} and g L p (Ω,μ){0}.

Remark 4.3 In the case where n=1 in Corollary 4.7, the coefficients of the upper and lower bounds of f g 1 are symmetric as follows:

f p g p ( 1 1 p | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 2 2 + ( 1 p 1 p ) | f | 1 / 2 f p 1 / 2 ( | f | ( p 1 ) / 2 f p ( p 1 ) / 2 | g | 1 / 2 g p 1 / 2 ) 2 2 ) f g 1 f p g p ( 1 1 p | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 2 2 ( 1 p 1 p ) | g | p / 4 g p p / 4 ( | f | p / 4 f p p / 4 | g | p / 4 g p p / 4 ) 2 2 ) .

Remark 4.4 Inequalities (4.8) improve the Aldaz stability version of the Hölder inequality [1]

f p g p ( 1 1 p | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 2 2 ) f g 1 f p g p ( 1 1 p | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 2 2 ) .
(4.9)

As Aldaz observed, (4.9) become

f p g p ( 1 2 p ) f g 1 =0 f p g p ( 1 2 p )

if suppfsuppg=. In this respect, Corollary 4.7 is sharp since both sides of the inequalities in (4.8) vanish as follows:

f p g p ( 1 1 p | f | p / 2 f p p / 2 | g | p / 2 g p p / 2 2 2 + ( 1 p 1 p ) | f | 1 / 2 f p 1 / 2 ( | f | ( p 1 ) / 2 f p ( p 1 ) / 2 | g | 1 / 2 g p 1 / 2 ) 2 2 ) = f p g p ( 1 2 p + 1 p 1 p ) = 0 , f p g p ( 1 1 p j = 1 n 2 j 1 | g | p ( 1 / 2 2 j ) g p p ( 1 / 2 2 j ) ( | f | p 2 j f p p 2 j | g | p 2 j g p p 2 j ) 2 2 p 2 n p | g | p ( 1 / 2 2 n 1 ) g p p ( 1 / 2 2 n 1 ) ( | f | p 2 n 1 f p p 2 n 1 | g | p 2 n 1 g p p 2 n 1 ) 2 2 ) = f p g p ( 1 2 p 1 p j = 2 n 2 j 1 p 2 n p ) = 0 .

In addition, (4.8) coincides with the polarization identity

( | f | , | g | ) = f 2 g 2 ( 1 1 2 | f | f 2 | g | g 2 2 2 )

when p=2, where (,) is the standard L 2 inner product.

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Acknowledgements

The authors are grateful to the referees for important remarks and suggestions.

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Correspondence to Kazumasa Fujiwara.

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Fujiwara, K., Ozawa, T. Stability of the Young and Hölder inequalities. J Inequal Appl 2014, 162 (2014) doi:10.1186/1029-242X-2014-162

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Keywords

  • Young inequality
  • Hölder inequality