- Research Article
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On Sharp Triangle Inequalities in Banach Spaces II
Journal of Inequalities and Applications volume 2010, Article number: 323609 (2010)
Abstract
Sharp triangle inequality and its reverse in Banach spaces were recently showed by Mitani et al. (2007). In this paper, we present equality attainedness for these inequalities in strictly convex Banach spaces.
1. Introduction
In recent years, the triangle inequality and its reverse inequality have been treated in [1–5] (see also [6, 7]).Kato et al.[8] presented the following sharp triangle inequality and its reverse inequality with elements in a Banach space
.
Theorem 1.1 (see [8]).
For all nonzero elements in a Banach space
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ1_HTML.gif)
These inequalities are useful to treat geometrical structure of Banach spaces, such as uniform non--ness (see [8]). Moreover,Hsu et al.[9] presented these inequalities for strongly integrable functions with values in a Banach space.
Mitani et al.[10] showed the following inequalities which are sharper than Inequality (1.1) in Theorem 1.1.
Theorem 1.2 (see [10]).
For all nonzero elements in a Banach space
with
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ2_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ3_HTML.gif)
where .
In this paper we first present a simpler proof of Theorem 1.2. To do this we consider the case as follows.
Theorem 1.3.
For all nonzero elements in a Banach space
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ5_HTML.gif)
where .
From this result we can easily obtain Theorem 1.2.
Moreover we consider equality attainedness for sharp triangle inequality and its reverse inequality in strictly convex Banach spaces. Namely, we characterize equality attainedness of Inequalities (1.4) and (1.5) in Theorem 1.3.
2. Simple Proofs of Theorems 1.2 and 1.3
Proof of Theorem 1.3.
According to Theorem 1.1 Inequalities (1.4) and (1.5) hold for the case (cf. [3]). Therefore let
. We first prove (1.4) by the induction. Assume that (1.4) holds true for all
elements in
. Let
be any
elements in
with
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ6_HTML.gif)
for all positive numbers with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ7_HTML.gif)
and . By assumption,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ8_HTML.gif)
holds, where . Since
, from (2.2) and (2.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ9_HTML.gif)
and hence (1.4). Thus (1.4) holds true for all finite elements in .
Next we show Inequality (1.5). Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ10_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ11_HTML.gif)
and Applying Inequality (1.4) to
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ12_HTML.gif)
where . Thus we obtain (1.5). This completes the proof.
Proof of Theorem 1.2.
Let be any nonzero elements in
with
. For all positive numbers
with
let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ13_HTML.gif)
Then . Applying Theorem 1.3 to
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ14_HTML.gif)
where for all positive numbers
with
. As
, we have Inequalities (1.4) and (1.5).
3. Equality Attainedness in a Strictly Convex Banach Space
In this section we consider equality attainedness for sharp triangle inequality and its reverse inequality in a strictly convex Banach space. Kato et al. in [8] showed the following.
Theorem 3.1 (see [8]).
Let be a strictly convex Banach space and
nonzero elements in
. Let
and
. Let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ15_HTML.gif)
if and only if either
(a)
or
(b)
Theorem 3.2 (see [8]).
Let be a strictly convex Banach space and
nonzero elements in
. Let
and
. Let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ16_HTML.gif)
if and only if either
(a)
or
(b)
We present equality attainedness for (1.4) and (1.5) in Theorem 1.2. The following lemma given in [8] is quite powerful.
Lemma 3.3 (see [8]).
Let be a strictly convex Banach space. Let
be nonzero elements in
. Then the following are equivalent.
(i) holds for any positive numbers
.
(ii) holds for some positive numbers
.
(iii).
Theorem 3.4.
Let be a strictly convex Banach space and
nonzero elements in
with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ17_HTML.gif)
if and only if there exists a real number with
satisfying
.
Proof.
Assume that (3.3) is true. By Theorem 3.1, the equality (3.3) is equivalent to Equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ18_HTML.gif)
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ19_HTML.gif)
Then . Since
, we obtain
. Conversely, if
where
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ20_HTML.gif)
By , we have (3.4). Thus we get (3.3).
Next we consider the case .
Theorem 3.5.
Let be a strictly convex Banach space and
nonzero elements in
with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ21_HTML.gif)
if and only if there exist with
satisfying one of the following conditions:
(a)
(b)
Proof.
Assume that (3.7) is true. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ22_HTML.gif)
Then and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ23_HTML.gif)
Note that . As in the proof of Theorem 1.2 given in [10], we have (3.7) if and only if we have the equalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ25_HTML.gif)
By Theorem 3.4, Equality (3.11) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ26_HTML.gif)
for some with
. By (3.8) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ27_HTML.gif)
for some . On the other hand, by Lemma 3.3, Equality (3.10) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ28_HTML.gif)
Hence, by using (3.8), (3.12), and (3.13) we have for some real number
. Since
, we have
. We consider the following two cases
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_IEq99_HTML.gif)
.
Equality (3.14) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ29_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ30_HTML.gif)
By and
, Equality (3.16) is valid for all real numbers
with
.
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_IEq104_HTML.gif)
.
Equality (3.14) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ31_HTML.gif)
So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ32_HTML.gif)
Hence . Thus (
) holds.
Conversely, assume that there exist with
satisfying one of the conditions
and
. Then it is clear that (3.7) holds. Thus we obtain (
).
Moreover we consider general cases. For each with
, we put
For
and
, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ33_HTML.gif)
For a finite set , the cardinal number of
is denoted by
.
Lemma 3.6.
Let . If
for all
with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ34_HTML.gif)
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ35_HTML.gif)
where and
. By the assumption, we have
. We first show
for all
with
. It is clear that
. Assume that
for all
with
. We will show
. Suppose that
. By
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ36_HTML.gif)
Hence we have which is a contradiction. Therefore we have
. Namely,
for all
with
. From this result, we obtain
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ37_HTML.gif)
Theorem 3.7.
Let be a strictly convex Banach space and
nonzero elements in
with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ38_HTML.gif)
if and only if there exists with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ40_HTML.gif)
for every with
.
Proof.
(): According to Theorems 3.4 and 3.5, Theorem 3.7 is valid for the cases
. Therefore let
. We will prove Theorem 3.7 by the induction. Assume that this theorem holds true for all nonzero elements in
less than
. Let
and assume that Equality (3.24) holds. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ41_HTML.gif)
for positive integer with
. As in the proof of Theorem 1.3, Equality (3.24) holds if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ43_HTML.gif)
hold, where . Hence, by assumption, there exists
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ44_HTML.gif)
Since by Lemma 3.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ45_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ46_HTML.gif)
by the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ47_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ48_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ49_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ50_HTML.gif)
for all with
. By Lemma 3.3, Equality (3.28) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ51_HTML.gif)
Hence there exists such that
. Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ52_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ53_HTML.gif)
Since , we have from (3.37) and (3.38),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ54_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ55_HTML.gif)
If , then it is clear that
.
If , then, by (3.41), we have
. Hence
. Thus we obtain
.
(): Let
with
satisfying (3.25) and (3.26), and let
. From (3.25) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ56_HTML.gif)
By Lemma 3.6,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ57_HTML.gif)
Let , where
. From (3.26) and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ58_HTML.gif)
Thus we obtain (3.24). This completes the proof.
In what follows, we characterize the equality condition of Inequality (1.3) in Theorem 1.2. For and positive integer
with
we define
, and
Lemma 3.8.
Let with
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ59_HTML.gif)
for all positive integers with
, then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ60_HTML.gif)
Proof.
Let and
, where
and
. As in the proof of Lemma 3.6, we have
for all
. So
for all
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ61_HTML.gif)
This completes the proof.
Theorem 3.9.
Let be a strictly convex Banach space and
nonzero elements in
with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ62_HTML.gif)
holds if and only if there exists with
,
for all positive integers
with
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ63_HTML.gif)
for all positive integers with
and
.
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_IEq212_HTML.gif)
: Let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ64_HTML.gif)
For positive integers with
we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ65_HTML.gif)
Note that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ66_HTML.gif)
Then Equality (3.48) holds if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ67_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ68_HTML.gif)
Thus, by the equality condition of sharp triangle inequality with elements, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ69_HTML.gif)
for all positive integers with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ70_HTML.gif)
for all positive integers with
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ71_HTML.gif)
for each with
, we have
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ72_HTML.gif)
Note that and
. By (3.53),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ73_HTML.gif)
Hence there exists such that
. We also have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ74_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ75_HTML.gif)
From (3.3), we obtain . Thus we have (
).
(): Assume that there exists
with
,
for all positive integers
with
satisfying (3.49) for all positive integers
with
and
. By Theorem 3.7, we have (3.54). From the assumption,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ76_HTML.gif)
By and Lemma 3.8, we obtain (3.53). Thus we have (
). This completes the proof.
If , then we have the following corollary.
Corollary 3.10.
Let be a strictly convex Banach space and
nonzero elements in
with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ77_HTML.gif)
if and only if there exists a real number with
such that
.
If , then we have the following corollary.
Corollary 3.11.
Let be a strictly convex Banach space and
nonzero elements in
with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ78_HTML.gif)
if and only if there exist with
such that
and
.
4. Remark
In this section we consider equality attainedness for sharp triangle inequality in a more general case, that is, the case without the assumption that . Let us consider the case
.
Proposition 4.1.
Let be a strictly convex Banach space, and
nonzero elements in
.
(i)If , then the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ79_HTML.gif)
always holds.
(ii)If , then the equality (4.1) holds if and only if there exists a real number
satisfying
and
.
(iii)If , then the equality (4.1) holds if and only if there exists a real number
satisfying
and
.
Proof.
-
(i)
Is clear.
-
(ii)
Assume that (4.1) holds. By
, (4.1) implies
(4.2)
From Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ81_HTML.gif)
Hence for some
. The following
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ82_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ83_HTML.gif)
Hence and so
The converse is clear.
-
(iii)
Assume that (4.1) holds. Put
and
. As in the proof of Theorem 3.5, we have
(4.6)
Since , we have
for some
. The following
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ85_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F323609/MediaObjects/13660_2009_Article_2120_Equ86_HTML.gif)
Hence . The converse is clear.
Conjecture 1.
What is the necessary and sufficient condition when Equality (3.24) (resp. Equality (3.48)) holds for elements
with
in Theorem 3.7 (resp. Theorem 3.9)?
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Acknowledgment
The second author was supported in part by Grants-in-Aid for Scientific Research (no. 20540158), Japan Society for the Promotion of Science.
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Mitani, KI., Saito, KS. On Sharp Triangle Inequalities in Banach Spaces II. J Inequal Appl 2010, 323609 (2010). https://doi.org/10.1155/2010/323609
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DOI: https://doi.org/10.1155/2010/323609