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.