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.
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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:
where and .
The standard Young inequality is described as
which may be used without particular comments. The standard Hölder inequality follows from (1.2) and the equality
for all and , where is the Banach space of q th integrable functions on a measure space with the norm , , and is the dual exponent of p defined by .
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 . In Section 2, we reexamine the multiplication formula on [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 in [1–4]. In Section 4, we give dyadic refinements of the multiplication formulae on 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 [1–7] and the references therein.
We close the introduction by giving some notation to be used in this paper. For we denote by and their minimum and maximum, respectively.
2 Multiplication formulae
In this section, we revisit the original multiplication formula on [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 . Then the equality
holds for all .
Proof The proposition follows from the equality
□
Corollary 2.2 Let θ and σ satisfy . Then the equality
holds for all .
Proposition 2.3 Let θ and σ satisfy . Then the equality
holds for all .
Remark 2.1 Equality (2.3) is regarded as a dual formula for in the sense that .
Proof of Proposition 2.3
□
Corollary 2.4 Let θ and σ satisfy . Then the equality
holds for all .
Remark 2.2 Propositions 2.1 and 2.3 are equivalent. In fact, it follows from the reciprocal formula and Proposition 2.1 that
which is precisely (2.3). Conversely, given θ and σ with , , we put and . Then we have , , , and . By the reciprocal formula and Proposition 2.3, we have
which is precisely (2.1).
Proposition 2.5 (Aldaz [1], Kichenassamy [2])
Let . Then the inequalities
hold for all .
Proof Though the first inequality of (2.5) is shown in [2], we show the inequalities in (2.5) for completeness. In the case , we use Corollaries 2.2 and 2.4 with to obtain
In the case , we apply (2.6) with θ replaced by to obtain
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 . For that purpose, we study the upper and lower bound estimates in terms of majorant and minorant in the form
for all . We introduce four couples of the bounds as follows:
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 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 . Then the inequalities
hold for all .
Proof By homogeneity, (3.1) follows from the inequality
for all and any θ and σ with . Inequality (3.3) follows from
Although some inequalities in (3.2) are proved in [4, 8], we prove (3.2) for completeness. By the integral representations [4, 8]
we have
Then it suffices to prove that
The last two inequalities are equivalent and follow from
for all . □
Proposition 3.2 Let and let
Then the following inequalities hold for all :
Remark 3.2 Since , satisfies
for all θ with . Proposition 3.2 shows that is better than in a neighborhood of the diagonal in the quarter plane .
Proof of Proposition 3.2 It is sufficient to show inequalities (3.4) and (3.5) with . We have
Moreover, is equivalent to the equation
Since the ratio of satisfying (3.6) with given θ is uniquely determined, inequalities (3.4) and (3.5) follow. □
To compare and , we prepare Lambert’s W function, which is defined as the inverse function of . For details, see [8].
Proposition 3.3 Let and let
where and are understood to be
Then the following inequalities hold for any :
Remark 3.3 Since , satisfies for . In the proof below, we see that if . Proposition 3.3 shows that is better than in a neighborhood of the diagonal in the quarter plane .
Proof of Proposition 3.3 Let satisfy . The magnitude correlation of and coincides with that of
and
Let . We have since
which is rewritten as
and, moreover,
In addition,
Then inequalities (3.7) and (3.8) follow from the Table 1. □
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 and their applications. By the reciprocal formula , it is important to describe the formation of the remainder function as and with the principal terms and . For that purpose, we utilize dyadic decomposition.
Proposition 4.1 Let θ satisfy with an integer . Then the equality
holds for all .
Proof We apply Corollary 2.2 with to obtain
for any j with . Then (4.1) follows immediately. □
Proposition 4.2 Let θ satisfy with an integer . Then the equality
holds for all .
Proof We apply Corollary 2.4 with to obtain
Then (4.2) follows by applying Proposition 4.1 to the last term on the right-hand side of (4.3) with . □
Corollary 4.3 Let . Then the inequalities
hold for all .
Corollary 4.4 Let . Then the inequalities
hold for all .
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 , while (4.4) become an equality when and (4.5) become an equality when .
We are now in a position to apply the equalities above to Hölder type inequalities.
Theorem 4.5 Let p satisfy and let m and n be unique integers satisfying
Then the equalities
hold for all and .
Proof The theorem follows from (1.3) and Propositions 4.1 and 4.2 with , , . □
Corollary 4.6 Let p, m, n be as in Theorem 4.5. Then the inequalities
hold for all and .
Proof The required inequalities follow from Theorem 4.5 and Proposition 2.5. □
Corollary 4.7 Let with a positive integer n. Then the inequalities
hold for all and .
Remark 4.3 In the case where in Corollary 4.7, the coefficients of the upper and lower bounds of are symmetric as follows:
Remark 4.4 Inequalities (4.8) improve the Aldaz stability version of the Hölder inequality [1]
As Aldaz observed, (4.9) become
if . In this respect, Corollary 4.7 is sharp since both sides of the inequalities in (4.8) vanish as follows:
In addition, (4.8) coincides with the polarization identity
when , where is the standard inner product.
References
Aldaz JM: A stability version of Hölder’s inequality. J. Math. Anal. Appl. 2008, 343: 842-852. 10.1016/j.jmaa.2008.01.104
Kichenassamy S: Improving Hölder’s inequality. Houst. J. Math. 2010, 36: 303-312.
Fujiwara K, Ozawa T: Exact remainder formula for the Young inequality and applications. Int. J. Math. Anal. 2013, 7: 2733-2735.
Hu X-L: An extension of Young’s inequality and its application. Appl. Math. Comput. 2013, 219: 6393-6399. 10.1016/j.amc.2013.01.001
Furuichi S, Minculete N: Alternative reverse inequalities for Young’s inequality. J. Math. Inequal. 2011, 5: 595-600.
Gao X, Gao M, Shang X: A refinement of Hölder’s inequality and applications. JIPAM. J. Inequal. Pure Appl. Math. 2007., 8: Article ID 44
Pečarić J, Šimić V: A note on the Hölder inequality. JIPAM. J. Inequal. Pure Appl. Math. 2006., 7: Article ID 176
Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE: On the Lambert W function. Adv. Comput. Math. 1996, 5: 329-359. 10.1007/BF02124750
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The authors are grateful to the referees for important remarks and suggestions.
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Fujiwara, K., Ozawa, T. Stability of the Young and Hölder inequalities. J Inequal Appl 2014, 162 (2014). https://doi.org/10.1186/1029-242X-2014-162
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DOI: https://doi.org/10.1186/1029-242X-2014-162