Abstract
Let be a concave function with . There is a corresponding map for which the inverse Minkowski inequality holds. Several properties of that map are obtained. Also, we consider the Beckenbach-Dresher type inequality connected with ψ-direct sums of Banach spaces and of ordered spaces. In the last section we investigate the properties of functions ψω,qand ∥.∥ω,q, (0 < ω < 1, q < 1) related to the Lorentz sequence space. Other posibilities for parameters ω and q are considered, the inverse Holder inequalities and more variants of the Beckenbach-Dresher inequalities are obtained.
2000 MSC: Primary 26D15; Secondary 46B99.
Similar content being viewed by others
1 Preliminaries
In the fifties of the previous century the following result was obtained:
Let 1 ≤ p ≤ 2 and x i , y i > 0, i = 1, ..., n. Then
The above-mentioned discrete inequality was given by Beckenbach [1], and the integral version is due to Dresher [2] (see also [3]). From that time, some generalizations of the Beckenbach-Dresher inequality (1) have appeared. Here, we are pointing out articles of Pečarić and Beesack [4], Petree and Persson [5], Persson [6] and Varošanec [7], where the reader can find related literature about this inequality Here we consider inequalities of Beckenbach-Dresher type in more general structures, namely in ψ-direct sums.
In this article we follow definitions and notations from the paper [8]. Let Ψ denote the family of all convex functions ψ on [0, 1] with ψ(0) = ψ(1) = 1 satisfying
It is known (see [9]), that Ψ is in one-to-one correspondence with the set N a of all absolute and normalized norms on C2, i.e., such that
Namely, if ∥.∥ ∈ N a and ψ(t) = ∥(1 - t, t)∥, then ψ ∈ Ψ. Conversely, if ψ ∈ Ψ, then
is a norm and ∥.∥ ψ ∈ N a .
Some examples of convex functions and the corresponding norms are the following:
Example 1. The convex functions, 1 ≤ p < ∞, 0 ≤ t ≤ 1, correspond to l p -norms of C2. For p = ∞, the function ψ∞(t) = max{t, 1 - t}, 0 ≤ t ≤ 1, corresponds to the norm l∞.
Example 2. Let
where {x*, y*} is the non-increasing rearrangement of {|x|, |y|}. If 0 < ω < 1, 1 ≤ q, then ∥.∥ ω, q is a norm of the two-dimensional Lorentz sequence space d(2)(ω, q), it belongs to N a and its dual norm was computed recently in[10]. The corresponding function from Ψ is
If, then we get a classical Lorentz l p,q -norm.
Example 3. For α, 1/2 ≤ α ≤ 1, let us define the following function
ψα ∈ Ψ and the corresponding norm is
Example 4. Let 1 ≤ q < p ≤ ∞ and. Let ψp,q,λ= max{ψ p , λψ q }. Then the corresponding norm is ∥.∥p,q,λ= max{∥.∥ p , λ∥.∥ q }.
Example 5. Let 1/2 ≤ β ≤ 1 and let ψ β (t) = max{1 - t, t, β} (note that neither ψ β ≥ ψ2nor ψ β ≤ ψ2.). The corresponding norm is
For ψ ∈ Ψ let denote the dual of the norm ∥.∥ ψ . From [11] we have that is an absolute normalized norm and the corresponding convex function ψ* ∈ Ψ is
for t with 0 ≤ t ≤ 1. For the norms ∥.∥ ψ and we have the following Hölder-type inequality:
where x1, x2, y1, y2 ∈ C.
Since the proof of Beckenbach-Dresher inequality can be obtained as an application of the Minkowski inequality the Holder inequality and its inverse inequalities in different cases, we are going to see what kind of such inequalities we could prove using some ideas of ψ-direct sums. In the following section we consider a family of concave functions . We prove some properties of concave functions and the inverse Minkowski inequality Using these results and combining with the known results about the family Ψ and normalized absolute norms we obtain a variant of the Beckenbach-Dresher inequality related to those norms. In the third section we are considering Ψ-direct sums of Banach and ordered spaces. Finally, the last section is devoted to the two-dimensional Lorentz sequence space and its variants. There we obtain several inequalites of the Hölder type.
2 A family of concave functions and a generalization of the Beckenbach-Dresher inequality
Let denotes the family of all concave functions on [0, 1] with . Let us define the map on C2 by
As it is proved in the next proposition for this map the inverse Minkowski inequality holds. For that reason we call it a pseudo-norm.
Proposition 6. Letbe a concave function on [0, 1]. Then the inverse Minkowski inequality holds, i.e. for u, v, z, w ∈ C the following is valid:
Proof. Using concavity of the function and the equality
we get that
The proof is complete.
Our next result reads:
Proposition 7. Letbe a concave function on [0, 1], . Thenis non-increasing on (0, 1] andis non-decreasing on [0, 1). Ifthen the words non-increasing and non-decreasing are replaced by decreasing and increasing, correspondingly.
Moreover, if 0 ≤ p ≤ r, 0 ≤ q ≤ s, we have that
Proof. Let 0 < s < t < 1.
-
(1)
Put p = t/s and write where . Then, by concavity of , we get that
Hence , i.e. . If , then the inequality is strict.
-
(2)
Let now . Then . By concavity of we find that
Hence i.e. . If , then the inequality is strict.
To prove the monotonity property we proceed like in [8]. Firstly if 0 ≤ p ≤ r, 0 ≤ q ≤ s, then
Using the fact that is non-increasing and is non-decreasing we obtain that
The proof is complete.
Let . Denote
for 0 ≤ t ≤ 1. The corresponding map is defined by (3). Using similar arguments as in [10] we can prove the following:
Proposition 8. Letbe symmetric with respect to t = 1/2. Then ψ*is symmetric with respect to. Moreover,
for all t with.
The following inverse Hölder-type inequality holds:
Proposition 9. Let x1, x2, y1, y2 ∈ C. If , then
Proof. From the definition of the function ψ* we get that
Putting in that inequality and using formula (3) we get (5).
Example 10. Let 0 < p < 1. The function ψ p belongs toand the related function ψ*is ψ q , where. In this case inequality (5) has a form
which is the reversed Hölder inequality in the simplest form, see e.g., [12, p. 99].
The following result is a variant of the Beckenbach-Dresher inequality, obtained by using the inverse Holder inequality of the concrete space l1/u, the Minkowski inequality for the norm ∥.∥ ψ and the inverse Minkowski inequality (4).
Theorem 11. Let. Letwhereand let.
If u ≥ 1, then
The inequality holds also when. If, then the inequality holds in the opposite direction.
Proof. To prove this inequality in the first case we use the Minkowski inequality for the norm ∥.∥ ψ , the inverse Minkowski inequality (4) for the pseudo-norm and the inverse Hölder inequality for the l1/u-norm:
Remark 12. Note that ifwe get the classical Beckenbach inequality (1) for n = 2.
3 Ψ-Direct sums of spaces and some more generalizations of the Beckenbach-Dresher inequality
The ψ-direct sum X ⊕ ψ Y of the Banach spaces X and Y is a direct sum X ⊕ ψ Y equipped with the norm ∥(x, y)∥ψ = ∥(∥x∥ X , ∥y∥Y)∥ ψ . This extends the notion of the l p -sum X ⊕ p Y. Recently various geometric properties of ψ- direct sums have been investigated by many authors [11, 13–18].
In the ψ-direct sum X ⊕ ψ Y of Banach spaces the Minkowski inequality holds, i.e., we have the following:
Let A and C be Banach spaces and let . Then
Let us improve that idea to the sum of ordered spaces.
Let B and D be ordered spaces equiped with pseudo-norms ∥.∥ B and ∥.∥ D and let . That means that in B and D the inverse Minkowski inequality holds, i.e.,
Let be a concave function from . Let us define . is called -direct sum of spaces B and D. Using monotonicity of the pseudo-norm generated by the function and the fact that the inverse Minkowski inequality holds for this pseudo-norm, we get the following estimates for the -direct sum:
So we have showed that when and the inverse Minkowski inequality holds in B and D, then this inequality holds also for the pseudo-norm of .
Our next result is the following inequalities of the Beckenbach-Dresher type:
Theorem 13. Letand A, C be Banach spaces, B, D ordered spaces such that the inverse Minkowski inequality holds. Let, for. If u ≥ 1, then
Proof. The proof is similar to the proof of Theorem 11 so we omit the details.
Remark 14. If u < 1, the analogue result can be considered.
If, then we get that
Note that if we take A = B = C = D = R and putthen we get the classical Beckenbach inequality (1) for n = 2.
A natural question arises : can we get some similar generalization of Beckenbach-Dresher inequality for n > 2?
We use the construction from [19]. Let Δ n be a set Δ n = {(s1, ..., sn-1) ∈ Rn-1: s1 + ... + sn-1≤ 1, s i ≥ 0, 1 ≤ i ≤ n - 1}. In [19] the authors considered the family Ψ n of all continuous convex functions ψ on Δ n which satisfy:
for k = 1, ..., n - 1. They showed that the family AN n of all absolute normalized norms on Cnand the family Ψ n are in one-to-one correspondence: if ∥.∥ ∈ AN n , then
belongs to Ψ n and for given ψ ∈ Ψ n the following norm ∥.∥ ψ belongs to AN n :
if (z1, ..., z n ) ≠ (0, ..., 0) and ∥(0, ..., 0)∥ ψ = 0.
The set is defined as a set of all positive continuous concave functions on Δ n , which satisfy condition (6).
Let us define the pseudo-norm by the formula given in (7). Consider for simplicity the case n = 3. If the function is concave it is not difficult to prove using the same idea as in Proposition 7, that, for λ > 1, it yields that
In the same way as it was done in Propositions 6 and 7 we can prove monotonicity of the pseudonorm and the inverse Minkowski inequality
As above we could prove that if B1, ..., B n have inverse Minkowski property and , then would have inverse Minkowski property. So we can state the following generalization of the previous Theorem:
Theorem 15. Letand A1, A2, ..., A n be Banach spaces, B1, B2, ..., B n be ordered spaces in which the inverse Minkowski inequality hold. Let, wherefor i = 1, 2, ..., n. If u ≥ 1, then
Proof. The proof is completely similar as that before, so we leave out the details.
4 Norm of the Lorentz sequence space and its variants
Let us consider two-dimensional Lorentz sequence space d(2)(ω, q), where 0 < ω < 1, 1 ≤ q. It is R2 with the norm
The corresponding convex function is
The dual norm of d(2)(ω, q) is completely determined by Mitani and Saito in [10] by finding the corresponding function . Namely, if 0 < ω < 1 and 1 < q < ∞, then we have that
where . The dual norm is equal to
If 0 < ω < 1 and q = 1, then
and
If 0 < q < 1, ω > 1, then ψω,qis a concave function from and we have the inverse Minkowski inequality for the corresponding pseudo-norm. Indeed, if 0 ≤ t ≤ 1/2, then the function is increasing and concave; if 1/2 ≤ t ≤ 1, then it is decreasing and concave (which can be shown by finding the first and second derivatives). Let 0 ≤ t1 ≤ 1/2 ≤ t2 ≤ 1. Consider the line connecting the points (t1, ψω,q(t1)) and (t2, ψω,q(t2)). For the concavity it is enough to show that the graph of the function is above this line. In fact, this is the case, because if we consider for instance t1 ≤ t ≤ 1/2, then (because of concavity on this interval) the graph is above the line connecting the points (t1, ψω,q(t1)) and (1/2, ψω,q(1/2)).
Lemma 1 from [20] for the case 1 ≤ q < ∞, 0 < ω ≤ 1 asserts that
for all a ∈ R2. It is easy to obtain similar result for other posibilities of parameters q and ω. For example the following holds:
Lemma 16. If ω ≥ 1 and q > 0, then
for all a ∈ R2.
Our next result reads:
Theorem 17. Let a, b ∈ R2. Then the following inequality holds
where
Proof. If q ≥ 1, 0 < ω ≤ 1, then Kato and Maligranda proved that C = 1 in [20].
Let q ≥ 1, 1 ≤ ω. Using (9) and the Minkowski inequality for the norm ∥.∥ q we have that
Let 0 < q < 1, 0 < ω ≤ 1. Using inequality between means of order q and 1 and superadditivity of the function f(s) = sqwe have the following:
where a = (a1, a2) and b = (b1, b2).
Combining the inequality and (8) we get that
Finally, let 0 < q < 1, ω ≥ 1. Using the above-proved inequality and (9) we have that
The proof is complete.
Remark 18. For the second case we see that the quasi-norm constant C is less than or equal to. We will compare with the result from[21], Proposition 1, whereand the quasi-norm constant is. Sincewe see that quasi-norm constant C obtained in this theorem is strictly less than the known constant for that case.
Remark 19. As we already mentioned, the norm of dual space of the space d(2)(ω, q), q ≥ 1, 0 < ω ≤ 1, is given in[10].
In the next part of this section we calculate the function ψ *ω, q for 0 < q < 1 and 1 ≤ ω and the corresponding mapping . Using those results we obtain examples of the inverse Holder inequality, and get some new variations of the Beckenbach-Dresher inequality.
Proposition 20. If 0 < q < 1, 1 ≤ ω, then
where .
Proof. We consider the function
for fixed t. Let first . Here .
The derivative of f is
and f'(s0) = 0 when
If , then 0 ≤ s0 ≤ 1/2. Therefore it is easy to see that the function f attains its minimum at s = s0 i.e.,
If , then s0 ≥ 1/2. Hence the minimum of f is at s = 1/2 i.e., .
Having in mind the symmetry of the function we can end the proof.
In the previous Proposition we consider t ∈ (0, 1) since for q ∈ (0, 1) p is negative. But, it is easy to see that .
Proposition 21. If 0 < q ≤ 1, 1 ≤ ω, then
Proof. Let x* ≥ ωy*. Without loosing of generality, put x* = |x|. This means that |x| ≥ ω|y|, and then and
The case when x* = |y| is quite analogue. Let x* ≤ ωy*. Let for instance x* = |x| i.e., |y| ≤ |x| ≤ ω|y|. Then and
Using the inverse Hölder inequality we obtain the following Corollary:
Corollary 22. Let 0 < q ≤ 1, 1 ≤ ω, x1, x2, y1, y2 > 0.
If, we have that
If , we find that
Let f i , g i be as in Theorem 13, let be such that if ∥f1∥ ψ ≥ ∥f2∥ ψ and if ∥f1∥ ψ ≤ ∥f2∥ ψ . Moreover, let be such that if if . Hence .
We state the following new variant of the Beckenbach-Dresher inequality:
Theorem 23. Let u ≥ 1, 1 ≤ ω. Let, f i , g i be as in Theorem 13 and. Then
Proof. Using the Minkowski inequality for the norm ∥.∥ ψ , inverse Minkowski inequality for and inverse Hölder inequality i.e., previous corollary for :
Remark 24. Let u ≥ 1, 1 ≤ ω. Letand. Then
We get this by replacing x2, y2in the above corollary by. If we use the inequalitywe get that
Remark 25. For the casewe get another variant of the Beckenbach-Dresher inequality, namely
Remark 26. The casesandcan be treated in a similar way.
For the case q ≥ 1, 0 < ω ≤ 1, ∥.∥ω,qis a norm, so we have the Minkowski and the Hölder inequality (2).
For the case q ≤ 1, ω ≤ 1we have Minkowski inequality with constant , inverse Minkowski, and inverse Holder inequalities.
For the case q ≥ 1, 1 ≤ ω, we have Minkowski inequality with constant . The function ψω,qis not convex. It has relative minimums at the points t0 and t1. We could try to improve it constructing a convex function changing it on the interval (t0, t1) by replacing it by a constant equal to ψω,q(t0) = ψω,q(t1). The corresponding norm is a norm indeed, but some calculations show that actually this is not a new norm, but the norm .
For the case q ≤ 1, 0 < ω ≤ 1, we have the Minkowski inequality with constant . The function ψω,qis not concave. It has relative maximum at the points t0 and t1. We can improve it constructing a concave function, namely changing it on the interval (t0, t1) by replacing it by a constant equal to ψω,q(t0) = ψω,q(t1). The new function is
where , and
Remark 27. Analogous results connected to the function ψp,q,λare given in the article[22].
References
Beckenbach EF: A class of mean-value functions. Am Math Mon 1950, 57: 1–6. 10.2307/2305163
Dresher M: Moment spaces and inequalities. Duke Math J 1953, 20: 261–271. 10.1215/S0012-7094-53-02026-2
Beckenbach EF, Bellman R: Inequalities. Springer, Berlin; 1961.
Pečarić JE, Beesack PR: On Jessen's inequality for convex functions II. J Math Anal Appl 1986, 118: 125–144. 10.1016/0022-247X(86)90296-9
Peetre J, Persson LE: A general Beckenbach's inequality with applications. In Function Spaces, Differential Operators and Nonlinear Analysis. Pitman Res Notes Math Ser 1989, 211: 125–139.
Persson LE: Generalizations of some classical inequalities with applications. Teubner Texte zur Mathematik 1991, 119: 127–148.
Varošanec S: The generalized Beckenbach inequality and related results. Banach J Math Anal 2010, 4(1):13–20.
Saito KS, Kato K, Takahashi Y: Von Neumann-Jordan constant of absolute normalized norms on C2. J Math Anal Appl 2000, 244: 515–532. 10.1006/jmaa.2000.6727
Bonsall FF, Duncan J: Numerical Ranges II, London Math. In Soc Lecture Note Ser. Volume 10. Cambridge University Press, Cambridge; 1973.
Mitani KI, Saito KS: Dual of two-dimensional Lorentz sequence spaces. Nonlinear Anal 2009, 71: 5238–5247. 10.1016/j.na.2009.04.007
Mitani KI, Oshiro S, Saito KS: Smoothness of Ψ-direct sums of Banach spaces. Math Inequal Appl 2005, 8: 147–157.
Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis. Kluwer Acad. Publishers, Dordrecht; 1993.
Kato M, Saito KS, Tamura T: On ψ -direct sums of Banach spaces and convexity. J Aust Math Soc 2003, 75: 413–422. 10.1017/S1446788700008193
Mitani KI, Saito KS: A note on geometrical properties of Banach spaces using ψ -direct sums. J Math Anal Appl 2007, 327: 898–907. 10.1016/j.jmaa.2006.04.059
Saito KS, Kato M: Uniform convexity of ψ -direct sums of Banach spaces. J Math Anal Appl 2003, 277: 1–11. 10.1016/S0022-247X(02)00282-2
Takahashi Y, Kato M, Saito KS: Strict convexity of absolute norms on C2and direct sums of Banach spaces. J Inequal Appl 2002, 7: 179–186.
Dowling PN: On convexity properties of ψ -direct sums of Banach spaces. J Math Anal Appl 2003, 288: 540–543. 10.1016/j.jmaa.2003.09.011
Dhompongsa S, Kaewkhao A, Saejung S: Uniform smoothness and U-convexity of ψ -direct sums. J Nonlinear Convex Anal 2005, 6: 327–338.
Saito KS, Kato M, Takahashi Y: Absolute norms on Cn. J Math Anal Appl 2000, 252: 879–905. 10.1006/jmaa.2000.7139
Kato M, Maligranda L: On James and Jordan-von Neumann constants of Lorentz sequences spaces. J Math Anal Appl 2001, 258: 457–465. 10.1006/jmaa.2000.7367
Kato M: On Lorentz spaces lp, q( E ). Hiroshima Math J 1976, 6: 73–93.
Nikolova L, Varošanec S: Refinements of Hölder's inequality derived from functions ψp, q,λand ϕp, q,λ. Ann Funct Anal 2011, 2: 72–83.
Acknowledgements
The first author thanks the University of Zagreb for kind hospitality and suport at a research visit in September 2010 when the main part of the article was done. The research of the first author was partially supported by the Sofia University SRF under contract no. 150/2011. The research of the third author was supported by the Ministry of Science, Education and Sports of the Republic of Croatia under grants 058-1170889-1050 and 037-1001677-2769.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors conceived of the study and carried out the proof. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Nikolova, L., Persson, LE. & Varošanec, S. The Beckenbach-Dresher inequality in the Ψ-direct sums of spaces and related results. J Inequal Appl 2012, 7 (2012). https://doi.org/10.1186/1029-242X-2012-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2012-7