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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 746309, 9 pages

http://dx.doi.org/10.1155/2014/746309

## -Uniform Convexity and -Uniform Smoothness of Absolute Normalized Norms on

Department of Basic Sciences, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

Received 19 September 2013; Accepted 15 November 2013; Published 16 February 2014

Academic Editor: Henryk Hudzik

Copyright © 2014 Tomonari Suzuki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We first prove characterizations of -uniform convexity and -uniform smoothness. We next give a formulation on absolute normalized norms on . Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is not -uniformly convex.

#### 1. Introduction

Throughout this paper, we denote by , , and the sets of positive integers, real numbers, and complex numbers, respectively.

Let be a *nontrivial* Banach space, which means a real Banach space with or a complex Banach space with . The *modulus of convexity* of is defined as
for , where the infimum can be taken over all with , , and . The *modulus of smoothness* of is defined as
for , where the supremum can be taken over all with and . It is obvious that . We know that if is a Hilbert space, then and .

We recall that is said to be *uniformly convex* if for all . Also, is said to be *uniformly smooth* if .

For , is called -uniformly convex if there exists satisfying for all . On the other hand, for , is called -uniformly smooth if there exists satisfying for all . It is obvious that -uniformly convex Banach spaces are uniformly convex, and -uniformly smooth Banach spaces are uniformly smooth. We also know that, for , spaces are -uniformly convex and -uniformly smooth. See [1–6] and others.

A norm on is said to be *absolute* if
for all and *normalized* if . The -norms are such examples:

Let be the family of all absolute normalized norms on . We let be the set of all convex functions on satisfying for . Bonsall and Duncan in [7] showed the following characterization of absolute normalized norms on . Namely, the set of all absolute normalized norms on is in one-to-one correspondence with . The correspondence is given by Indeed, for any , the norm on defined as belongs to and satisfies (8). Saito et al. in [8] extended this result to .

In this paper, we first prove characterizations of -uniform convexity and -uniform smoothness. We next give another formulation on absolute normalized norms on . Using these, we present some examples, one of which is a uniformly convex Banach space which is not -uniformly convex.

#### 2. Characterizations

In this section, we prove characterizations of -uniform convexity and -uniform smoothness.

Proposition 1. *Let be a Banach space and let . Then the following are equivalent: *(i) *is **-uniformly convex, *(ii).

*Proof. *We first assume that . Then for every , there exists a small such that . That is, is not -uniformly convex. Conversely, we next assume that is not -uniformly convex. That is, for every , there exists such that . Putting , we can define a sequence in such that . In the case of , without loss of generality, we may assume . We have
and hence . In the other case, there exists such that for all . Then since is nondecreasing, we have
for and hence . Therefore, for . This implies .

Proposition 2. *Let be a Banach space and let . Then the following are equivalent:*(i)* is **-uniformly smooth,*(ii).

*Proof. *We first assume that . Then for every , there exists a small such that . That is, is not -uniformly smooth. Conversely, we next assume that is not -uniformly smooth. That is, for every , there exists such that . Putting , we can define a sequence in such that . Then we have
Hence, because . Therefore, we obtain
This completes the proof.

We know that Hilbert spaces are 2-uniformly convex and 2-uniformly smooth Banach spaces. We can easily check this thing by Propositions 1 and 2.

#### 3. Convex Functions

In this section, we discuss properties of convex functions belonging to . We first note that functions belonging to are continuous and satisfy and for all .

Let . Then we define , , and as follows: for , for , and for . See [9] and others.

We know the following.

Lemma 3 (see [9, 10]). *Let . Then the following hold: *(i)*For with ,
* *holds.*(ii)*For with ,
* *holds.*(iii)*For ,
* *holds.*(iv)* holds.*(v)* and hold.*

*Remark 4. *(i)–(iii) are stated in [9]. (iv) follows from Theorem 24.1 in [9]. (v) is proved in [10].

Using Lemma 3, we can easily prove the following.

Lemma 5. *Let . Then the following hold:*(i)* for every ,*(ii)* for every .*

Lemma 6. *Let and with . Then
**
hold.*

The following lemma is used in Section 5.

Lemma 7. *Let and with . Then
**
holds.*

*Proof. *In the case of , we have
Using Lemma 6, we will prove this lemma in the other cases. In the case of , since , we have
In the case of , since , we have
This completes the proof.

We also know the following.

Lemma 8 (Bonsall and Duncan [7] page 37). *Let . Then the following hold: *(i)*the function is nonincreasing;*(ii)*the function is nondecreasing.*

The following lemma follows from Lemma 8.

Lemma 9. *Let and with . Then
**
hold.*

#### 4. Absolute Normalized Norms on

We denote by the set of nondecreasing functions from into satisfying . The following proposition says there are many absolute normalized norms on , and we can make many such norms easily.

Proposition 10. *Define a mapping from into by
**
for and , and define a mapping from into by
**
for and . Then a.e. and for all and .*

*Proof. *Fix and put . We will show . By Lemma 3, is nondecreasing, and . Hence for all . By the definition of , we have
This implies . Therefore, we have shown . Next, we fix and put . We will will . Since is nondecreasing, we have that is convex. It is obvious that . From the convexity of , for all . Since for , we have
for . Since for , we also have
for . Therefore . The remains are obvious.

We next discuss the convexity and smoothness. In [11], Takahashi et al. proved that is strictly convex if and only if is strictly convex. See also [8]. Using this fact, we can obtain the following.

Proposition 11. *Let . Then is strictly convex if and only if is injective.*

*Proof. *We assume that is strictly convex. Then is strictly convex. That is, for with , we have
Hence is injective. We can easily prove the converse implication.

In [10], Mitani et al. proved that is smooth if and only if is differentiable at any and and . Using this fact, we can prove the following.

Proposition 12. *Let . Then is smooth if and only if is surjective.*

*Proof. *We assume that is smooth. Then is differentiable at any and and . So and are obvious. We note that and . For , there exists with . From the above note, we have . From the differentiability, we obtain
That is, . Therefore we have shown is surjective. Conversely, we next assume that is surjective. We suppose that is not differentiable at some . Then we have . By Lemma 3, we have
This contradicts the surjectivity of . Hence, is differentiable at any . We next suppose that . Then by Lemma 3 again, we have
This is a contradiction. Hence, . We can similarly prove . Therefore, is smooth.

#### 5. Examples

In this section, we present examples of absolute normalized norms on satisfying that is uniformly convex and is not -uniformly convex. We also present examples of such norms satisfying that is uniformly smooth and is not -uniformly smooth. We note that, in finite dimensional Banach spaces, strict convexity and uniform convexity are equivalent. Smoothness and uniform smoothness are also equivalent.

Theorem 13. *Let and . Assume that there exist sequences and in such that for ,
**
Then is not -uniformly convex.*

*Proof. *Put . Without loss of generality, we may assume
for , and and converge to some number . We put
for . It is clear that for . Define sequences and in by
for . It is obvious . Then we have
Thus,
We put
By Lemma 9,
From this inequality and (46), holds. Using , we also have
Therefore, we obtain
We will show . Before showing it, we need some inequalities:
by Lemma 7. From (45) and (46), we have
These imply . So by Proposition 1, we obtain the desired result.

Corollary 14. *Let . Assume that is injective, is infinitely differentiable on the neighborhood of some , and
**
Then is uniformly convex and is not -uniformly convex for all .*

*Proof. *Put . By Proposition 11, since is injective, is strictly convex and hence it is uniformly convex. By the L’Hospital theorem, for with , we have
So, by Theorem 13, we have that is not -uniformly convex for every with . Therefore, we obtain the desired result.

It is well known that a function from into defined by for is strictly increasing on , infinitely differentiable and for all .

*Example 15. *Define by
for . Then is uniformly convex and not -uniformly convex for all . See Figure 1.

Theorem 16. *Let and . Assume that there exist a constant and sequences and in such that for ,
**
Then is not -uniformly smooth.*

*Proof. *Put . Without loss of generality, we may assume
for , and and converge to some number . We define a sequence by (37). Since
we may also assume that
for . We note that
because
for . Define sequences and in by
for . It is obvious that . We put by (41). We have
by Lemma 7. We note that . We will calculate and . We have
because
Hence,
for . Similarly, we obtain
and hence
for . Therefore, we obtain
From
and (59), we have
Hence we obtain . So by Proposition 2, we obtain the desired result.

Corollary 17. *Let be a bijective and strictly increasing function from into with . Assume that is infinitely differentiable on the neighborhood of some , and
**
Then and is uniformly smooth and is not -uniformly smooth for all .*

*Proof. *It is not difficult to check . Put and . By Proposition 12, since is surjective, is smooth and hence it is uniformly smooth. Fix . As in the proof of Corollary 14, we can prove . Since is strictly increasing, we have
Putting and , we have
We choose a strictly increasing sequence and a strictly decreasing sequence in satisfying for and . Then it is obvious that for and . We have
Thus, by Theorem 16, we have that is not -uniformly smooth. Since is arbitrary, is not -uniformly smooth for every .

*Example 18. *Define a function from onto by
for . Then is uniformly smooth and not -uniformly smooth for all . See Figure 1.

*Example 19. *Let be as in Example 15 and let be as in Example 18. Define a function from into by
for . Then is uniformly convex, uniformly smooth, not -uniformly convex for all , and not -uniformly smooth for all . See Figure 1.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The author is supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

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