# p-Uniform Convexity and q-Uniform Smoothness of Absolute Normalized Norms on ℂ2

Authors: Tomonari Suzuki
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We first prove characterizations ofp-uniform convexity and q-uniform
smoothness. We next give a formulation on absolute normalized norms on ℂ2. Using
these, we present some examples of Banach spaces. One of them is a uniformly convex
Banach space which is not p-uniformly convex.


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## Body

## 1. Introduction

Throughout this paper, we denote byℕ, ℝ, and ℂ the sets of positive integers, real numbers, and complex numbers, respectively.LetX be a nontrivial Banach space, which means a real Banach space with dim⁡X≥2 or a complex Banach space with dim⁡X≥1. The modulus of convexity of X is defined as
(1)δ(ε)=inf⁡(1-∥x+y∥2)
for ε∈[0,2], where the infimum can be taken over all x,y∈X with ∥x∥≤1, ∥y∥≤1, and ∥x-y∥≥ε. The modulus of smoothness of X is defined as
(2)ρ(τ)=sup⁡(∥x+τy∥+∥x-τy∥2-1)
for τ∈(0,∞), where the supremum can be taken over all x,y∈X with ∥x∥≤1 and ∥y∥≤1. It is obvious that ρ(τ)≤τ. We know that if X is a Hilbert space, then δ(ε)=1-1-ε2/4 and ρ(τ)=1+τ2-1.We recall thatX is said to be uniformly convex if δ(ε)>0 for all ε>0. Also, X is said to be uniformly smooth if lim⁡τ→+0⁡ρ(τ)/τ=0.Forp∈[2,∞), X is called p-uniformly convex if there exists C>0 satisfying
(3)δ(ε)≥Cεp
for all ε∈[0,2]. On the other hand, for q∈(1,2], X is called q-uniformly smooth if there exists K>0 satisfying
(4)ρ(τ)≤Kτq
for all τ∈(0,∞). It is obvious that p-uniformly convex Banach spaces are uniformly convex, and q-uniformly smooth Banach spaces are uniformly smooth. We also know that, for p∈(1,∞), Lp spaces are max⁡{2,p}-uniformly convex and min⁡{2,p}-uniformly smooth. See [1–6] and others.A norm∥·∥ on ℂ2 is said to be absolute if
(5)∥(x1,x2)∥=∥(|x1|,|x2|)∥
for all (x1,x2)∈ℂ2 and normalized if ∥(1,0)∥=∥(0,1)∥=1. The ℓp-norms ∥·∥p are such examples:
(6)∥(x1,x2)∥p={(|x1|p+|x2|p)1/p,if1≤p<∞,max⁡{|x1|,|x2|},ifp=∞.LetAN2 be the family of all absolute normalized norms on ℂ2. We let Ψ2 be the set of all convex functions ψ on [0,1] satisfying
(7)max⁡{1-t,t}≤ψ(t)≤1
for t∈[0,1]. Bonsall and Duncan in [7] showed the following characterization of absolute normalized norms on ℂ2. Namely, the set AN2 of all absolute normalized norms on ℂ2 is in one-to-one correspondence with Ψ2. The correspondence is given by
(8)ψ(t)=∥(1-t,t)∥fort∈[0,1].
Indeed, for any ψ∈Ψ2, the norm ∥·∥ψ on ℂ2 defined as
(9)∥(x1,x2)∥ψ={(|x1|+|x2|)×ψ(|x2||x1|+|x2|),if(x1,x2)≠(0,0),0,if(x1,x2)=(0,0)
belongs to AN2 and satisfies (8). Saito et al. in [8] extended this result to ℂn.In this paper, we first prove characterizations ofp-uniform convexity and q-uniform smoothness. We next give another formulation on absolute normalized norms on ℂ2. Using these, we present some examples, one of which is a uniformly convex Banach space which is not p-uniformly convex.


## 2. Characterizations

In this section, we prove characterizations ofp-uniform convexity and q-uniform smoothness.Proposition 1.
LetX be a Banach space and let p∈[2,∞). Then the following are equivalent: (i)
X isp-uniformly convex,(ii)
liminf⁡ε→+0⁡δ(ε)/εp>0.Proof.
We first assume thatliminf⁡ε→+0⁡δ(ε)/εp=0. Then for every C>0, there exists a small ε>0 such that δ(ε)/εp<C. That is, X is not p-uniformly convex. Conversely, we next assume that X is not p-uniformly convex. That is, for every C>0, there exists ε∈(0,2] such that δ(ε)<Cεp. Putting C=1/n, we can define a sequence {εn} in (0,2] such that δ(εn)/εnp<1/n. In the case of liminf⁡nεn=0, without loss of generality, we may assume lim⁡nεn=0. We have
(10)0≤liminf⁡ε→+0δ(ε)εp≤liminf⁡n→∞δ(εn)εnp≤lim⁡n→∞1n=0
and hence liminf⁡ε→+0δ(ε)/εp=0. In the other case, there exists ε0>0 such that ε0<εn for all n∈ℕ. Then since δ is nondecreasing, we have
(11)0≤δ(ε0)2p≤δ(εn)εnp<1n
for n∈ℕ and hence δ(ε0)=0. Therefore, δ(ε)=0 for ε∈[0,ε0]. This implies liminf⁡ε→+0δ(ε)/εp=0.Proposition 2.
LetX be a Banach space and let q∈(1,2]. Then the following are equivalent:(i)
Xisq-uniformly smooth,(ii)
limsup⁡τ→+0ρ(τ)/τq<∞.Proof.
We first assume thatlimsup⁡τ→+0ρ(τ)/τq=∞. Then for every K>0, there exists a small τ>0 such that ρ(τ)/τq>K. That is, X is not q-uniformly smooth. Conversely, we next assume that X is not q-uniformly smooth. That is, for every K>0, there exists τ>0 such that ρ(τ)>Kτq. Putting K=n, we can define a sequence {τn} in (0,∞) such that ρ(τn)/τnq>n. Then we have
(12)n<ρ(τn)τnq≤τnτnq=1τnq-1.
Hence, lim⁡nτn=0 because q-1>0. Therefore, we obtain
(13)limsup⁡τ→+0⁡ρ(τ)τq≥limsup⁡n→∞⁡ρ(τn)τnq≥lim⁡n→∞⁡n=∞.
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 Propositions1 and 2.


## 3. Convex Functions

In this section, we discuss properties of convex functions belonging toΨ2. We first note that functions ψ belonging to Ψ2 are continuous and satisfy ψ(0)=ψ(1)=1 and ψ(t)≥1/2 for all t∈[0,1].Letψ∈Ψ2. Then we define ψ-′, ψ+′, and ∂ψ as follows:
(14)ψ-′(s)=lim⁡t→s-0⁡ψ(t)-ψ(s)t-s
for s∈(0,1],
(15)ψ+′(s)=lim⁡t→s+0⁡ψ(t)-ψ(s)t-s
for s∈[0,1), and
(16)∂ψ(s)={a∈ℝ:ψ(t)≥ψ(s)+a(t-s)∀t∈[0,1]}
for s∈[0,1]. See [9] and others.We know the following.Lemma 3 (see [9, 10]).
Letψ∈Ψ2. Then the following hold: (i)
Fors,t,u∈[0,1] with 0≤s<t<u≤1,
(17)ψ(t)-ψ(s)t-s≤ψ(u)-ψ(s)u-s≤ψ(u)-ψ(t)u-tholds.(ii)
Fors,t,u∈[0,1] with 0≤s<t<u≤1,
(18)ψ+′(s)≤ψ(t)-ψ(s)t-s≤ψ-′(t)≤ψ+′(t)≤ψ(u)-ψ(t)u-t≤ψ-′(u)holds.(iii)
Fort∈[0,1],
(19)∂ψ(t)={(-∞,ψ+′(0)],ift=0,[ψ-′(t),ψ+′(t)],if0<t<1,[ψ-′(1),+∞),ift=1holds.(iv)
⋃‍{∂ψ(t):t∈[0,1]}=ℝ holds.(v)
-1≤ψ+′(0) and ψ-′(1)≤1 hold.Remark 4.
(i)–(iii) are stated in [9]. (iv) follows from Theorem  24.1 in [9]. (v) is proved in [10].Using Lemma3, we can easily prove the following.Lemma 5.
Letψ∈Ψ2. Then the following hold:(i)
ψ+′(t)≤(1-ψ(t))/(1-t) for every t∈[0,1),(ii)
ψ-′(t)≥(ψ(t)-1)/t for every t∈(0,1].Lemma 6.
Letψ∈Ψ2 and s,t,u∈[0,1] with s<t<u. Then
(20)-1≤ψ(t)-ψ(s)t-s≤1-ψ(t)1-t,ψ(t)-1t≤ψ(u)-ψ(t)u-t≤1
hold.The following lemma is used in Section5.Lemma 7.
Letψ∈Ψ2 and s,u∈[0,1] with s<u. Then
(21)u-s≤ψ(u)(1-2s)+ψ(s)(2u-1)≤2(u-s)
holds.Proof.
In the case ofs≤1/2≤u, we have
(22)u-s=12(1-2s)+12(2u-1)u-s≤ψ(u)(1-2s)+ψ(s)(2u-1)u-s≤(1-2s)+(2u-1)u-s=2(u-s).
Using Lemma 6, we will prove this lemma in the other cases. In the case of s>1/2, since 2ψ(s)-((ψ(s)-1)/s)(2s-1)≤2, we have
(23)u-s≤(2ψ(u)+1-2u)(u-s)u-s=2ψ(u)(u-s)-(u-s)(2u-1)u-s≤2ψ(u)(u-s)-(ψ(u)-ψ(s))(2u-1)u-s=ψ(u)(1-2s)+ψ(s)(2u-1)u-s=2ψ(s)(u-s)-(ψ(u)-ψ(s))(2s-1)u-s≤2ψ(s)(u-s)-ψ(s)-1s(u-s)(2s-1)u-s=(2ψ(s)-ψ(s)-1s(2s-1))(u-s)u-s≤2(u-s).
In the case of u<1/2, since 2ψ(u)-((1-ψ(u))/(1-u))(2u-1)≤2, we have
(24)u-s≤(2ψ(s)+2s-1)(u-s)u-s=2ψ(s)(u-s)+(u-s)(2s-1)u-s≤2ψ(s)(u-s)-(ψ(u)-ψ(s))(2s-1)u-s=ψ(u)(1-2s)+ψ(s)(2u-1)u-s=2ψ(u)(u-s)-(ψ(u)-ψ(s))(2u-1)u-s≤2ψ(u)(u-s)-1-ψ(u)1-u(u-s)(2u-1)u-s=(2ψ(u)-1-ψ(u)1-u(2u-1))(u-s)u-s≤2(u-s).
This completes the proof.We also know the following.Lemma 8 (Bonsall and Duncan [7] page 37).
Letψ∈Ψ2. Then the following hold: (i)
the functiont↦ψ(t)/t is nonincreasing;(ii)
the functiont↦ψ(t)/(1-t) is nondecreasing.The following lemma follows from Lemma8.Lemma 9.
Letψ∈Ψ2 and s,u∈[0,1] with s<u. Then
(25)sψ(s)≤uψ(u),1-sψ(s)≥1-uψ(u)
hold.


## 4. Absolute Normalized Norms onℂ2

We denote byΓ2 the set of nondecreasing functions γ from [0,1] into [-1,1] satisfying ∫01‍γ(s)ds=0. The following proposition says there are many absolute normalized norms on ℂ2, and we can make many such norms easily.Proposition 10.
Define a mappingD from Ψ2 into Γ2 by
(26)(Dψ)(t)={ψ+′(t),ift∈[0,1),ψ-′(t),ift=1
for ψ∈Ψ2 and t∈[0,1], and define a mapping S from Γ2 into Ψ2 by
(27)(Sγ)(t)=1+∫0t‍γ(s)ds
for γ∈Γ2 and t∈[0,1]. Then D∘Sγ=γ a.e. and S∘Dψ=ψ for all γ∈Γ2 and ψ∈Ψ2.Proof.
Fixψ∈Ψ2 and put γ=Dψ. We will show γ∈Γ2. By Lemma 3, γ is nondecreasing, -1≤ψ+′(0)=γ(0) and γ(1)=ψ-′(1)≤1. Hence γ(t)∈[-1,1] for all t∈[0,1]. By the definition of D, we have
(28)1=ψ(1)=ψ(0)+∫01‍γ(s)ds=1+∫01‍γ(s)ds.
This implies ∫01‍γ(s)ds=0. Therefore, we have shown γ∈Γ2. Next, we fix γ∈Γ2 and put ψ=Sγ. We will will Sγ∈Ψ2. Since γ is nondecreasing, we have that ψ is convex. It is obvious that ψ(0)=ψ(1)=1. From the convexity of ψ, ψ(t)≤1 for all t∈[0,1]. Since -1≤γ(t) for t∈[0,1], we have
(29)ψ(t)=1+∫0t‍γ(s)ds≥1+∫0t‍(-1)ds=1-t
for t∈[0,1]. Since γ(t)≤1 for t∈[0,1], we also have
(30)ψ(t)=1+∫0t‍γ(s)dsψ(t)=1+∫01‍γ(s)ds-∫t1‍γ(s)dsψ(t)=1-∫t1‍γ(s)ds≥1-∫t1‍1ds=t
for t∈[0,1]. Therefore ψ∈Ψ2. The remains are obvious.We next discuss the convexity and smoothness. In [11], Takahashi et al. proved that (ℂ2,∥·∥ψ) is strictly convex if and only if ψ is strictly convex. See also [8]. Using this fact, we can obtain the following.Proposition 11.
Letψ∈Ψ2. Then (ℂ2,∥·∥ψ) is strictly convex if and only if Dψ is injective.Proof.
We assume that(ℂ2,∥·∥ψ) is strictly convex. Then ψ is strictly convex. That is, for s,t,u∈[0,1] with 0≤s<t<u≤1, we have
(31)ψ+′(s)<ψ-′(t)≤ψ+′(t)<ψ-′(u).
Hence Dψ is injective. We can easily prove the converse implication.In [10], Mitani et al. proved that (ℂ2,∥·∥ψ) is smooth if and only if ψ is differentiable at any t∈(0,1) and ψ+′(0)=-1 and ψ-′(1)=1. Using this fact, we can prove the following.Proposition 12.
Letψ∈Ψ2. Then (ℂ2,∥·∥ψ) is smooth if and only if Dψ is surjective.Proof.
We assume that(ℂ2,∥·∥ψ) is smooth. Then ψ is differentiable at any t∈(0,1) and ψ+′(0)=-1 and ψ-′(1)=1. So (Dψ)(0)=-1 and (Dψ)(1)=1 are obvious. We note that ∂ψ(0)=(-∞,-1] and ∂ψ(1)=[1,+∞). For a∈(-1,1), there exists t∈[0,1] with a∈∂ψ(t). From the above note, we have t∈(0,1). From the differentiability, we obtain
(32)a∈∂ψ(t)=[ψ-′(t),ψ+′(t)]a∈∂ψ(t)={ψ′(t)}={ψ+′(t)}={(Dψ)(t)}.
That is, (Dψ)(t)=a. Therefore we have shown Dψ is surjective. Conversely, we next assume that Dψ is surjective. We suppose that ψ is not differentiable at some t∈(0,1). Then we have ψ-′(t)<ψ+′(t). By Lemma 3, we have
(33)(Dψ)([0,1])⊂[-1,1]∖(ψ-′(t),ψ+′(t))⫋[-1,1].
This contradicts the surjectivity of Dψ. Hence, ψ is differentiable at any t∈(0,1). We next suppose that -1<ψ+′(0). Then by Lemma 3 again, we have
(34)(Dψ)([0,1])⊂[-1,1]∖[-1,ψ+′(0))⫋[-1,1].
This is a contradiction. Hence, ψ+′(0)=-1. We can similarly prove ψ-′(1)=1. Therefore, (ℂ2,∥·∥ψ) is smooth.


## 5. Examples

In this section, we present examples of absolute normalized norms onℂ2 satisfying that (ℂ2,∥·∥ψ) is uniformly convex and is not p-uniformly convex. We also present examples of such norms satisfying that (ℂ2,∥·∥ψ) is uniformly smooth and is not q-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.
Letγ∈Γ2 and p∈[2,∞). Assume that there exist sequences {sn} and {un} in [0,1] such that sn<un for n∈ℕ,
(35)lim⁡n→∞(un-sn)=0,lim⁡n→∞γ(un)-γ(sn)(un-sn)p-1=0.
Then (ℂ2,∥·∥Sγ) is not p-uniformly convex.Proof.
Putψ=Sγ. Without loss of generality, we may assume
(36)γ(un)-γ(sn)(un-sn)p-1≤1n
for n∈ℕ, and {sn} and {un} converge to some number t0∈[0,1]. We put
(37)tn=(sn/ψ(sn))+(un/ψ(un))(1/ψ(sn))+(1/ψ(un))
for n∈ℕ. It is clear that sn<tn<un for n∈ℕ. Define sequences {xn} and {yn} in ℂ2 by
(38)xn=1ψ(sn)(1-sn,sn),yn=1ψ(un)(1-un,un)
for n∈ℕ. It is obvious ∥xn∥=∥yn∥=1. Then we have
(39)xn+yn=(1-snψ(sn)+1-unψ(un),snψ(sn)+unψ(un))xn+yn=(1ψ(sn)+1ψ(un))(1-tn,tn).
Thus,
(40)∥xn+yn∥=(1ψ(sn)+1ψ(un))ψ(tn).
We put
(41)vn=(un/ψ(un))-(sn/ψ(sn))((1-2sn)/ψ(sn))+((2un-1)/ψ(un)).
By Lemma 9,
(42)0≤unψ(un)-snψ(sn)≤1-2snψ(sn)+2un-1ψ(un).
From this inequality and (46), vn∈[0,1] holds. Using vn, we also have
(43)∥xn-yn∥=∥(1-snψ(sn)-1-unψ(un),snψ(sn)-unψ(un))∥∥xn-yn∥=∥(1-snψ(sn)-1-unψ(un),unψ(un)-snψ(sn))∥∥xn-yn∥=(1-2snψ(sn)+2un-1ψ(un))∥(1-vn,vn)∥∥xn-yn∥=(1-2snψ(sn)+2un-1ψ(un))ψ(vn).
Therefore, we obtain
(44)δ((1-2snψ(sn)+2un-1ψ(un))ψ(vn))≤1-12(1ψ(sn)+1ψ(un))ψ(tn).
We will show liminf⁡ε→+0δ(ε)/εp=0. Before showing it, we need some inequalities:
(45)2ψ(sn)ψ(un)-(ψ(un)+ψ(sn))ψ(tn)=ψ(sn)(ψ(un)-ψ(tn))-ψ(un)(ψ(tn)-ψ(sn))=ψ(sn)∫tnun‍γ(s)ds-ψ(un)∫sntn‍γ(s)ds≤ψ(sn)γ(un)(un-tn)-ψ(un)γ(sn)(tn-sn)=ψ(sn)γ(un)(un-(sn/ψ(sn))+(un/ψ(un))(1/ψ(sn))+(1/ψ(un)))-ψ(un)γ(sn)((sn/ψ(sn))+(un/ψ(un))(1/ψ(sn))+(1/ψ(un))-sn)=1(1/ψ(sn))+(1/ψ(un))(γ(un)-γ(sn))(un-sn)≤(γ(un)-γ(sn))(un-sn)≤1n(un-sn)p,(46)(1-2snψ(sn)+2un-1ψ(un))ψ(vn)=(ψ(un)(1-2sn)+ψ(sn)(2un-1))×ψ(vn)ψ(sn)ψ(un)≥(un-sn)ψ(vn)ψ(sn)ψ(un)≥12(un-sn)>0
by Lemma 7. From (45) and (46), we have
(47)δ((((1-2sn)/ψ(sn))+((2un-1)/ψ(un)))ψ(vn))((((1-2sn)/ψ(sn))+((2un-1)/ψ(un)))ψ(vn))p≤1-(1/2)((1/ψ(sn))+(1/ψ(un)))ψ(tn)((((1-2sn)/ψ(sn))+((2un-1)/ψ(un)))ψ(vn))p=12ψ(sn)ψ(un)×2ψ(sn)ψ(un)-(ψ(un)+ψ(sn))ψ(tn)((((1-2sn)/ψ(sn))+((2un-1)/ψ(un)))ψ(vn))p≤12ψ(sn)ψ(un)×(un-sn)pn((((1-2sn)/ψ(sn))+((2un-1)/ψ(un)))ψ(vn))p≤12nψ(sn)ψ(un)2p≤2n2p,limsup⁡n→∞(1-2snψ(sn)+2un-1ψ(un))ψ(vn)≤lim⁡n→∞(1-2snψ(sn)+2un-1ψ(un))=1-2t0ψ(t0)+2t0-1ψ(t0)=0.
These imply liminf⁡ε→+0δ(ε)/εp=0. So by Proposition 1, we obtain the desired result.Corollary 14.
Letγ∈Γ2. Assume that γ is injective, γ is infinitely differentiable on the neighborhood of some t0∈(0,1), and
(48)γ′(t0)=γ′′(t0)=γ′′′(t0)=⋯=0.
Then (ℂ2,∥·∥Sγ) is uniformly convex and is not p-uniformly convex for all p∈[2,∞).Proof.
Putψ=Sγ. By Proposition 11, since γ is injective, (ℂ2,∥·∥ψ) is strictly convex and hence it is uniformly convex. By the L’Hospital theorem, for n∈ℕ with n≥2, we have
(49)0=lim⁡u→t0+0γ(n-1)(u)(n-1)!0=lim⁡u→t0+0γ(n-2)(u)(n-1)!/1!(u-t0)0=lim⁡u→t0+0γ(n-3)(u)(n-1)!/2!(u-t0)2l0ll⋮0=lim⁡u→t0+0γ′(u)(n-1)(u-t0)n-20=lim⁡u→t0+0γ(u)-γ(t0)(u-t0)n-1.
So, by Theorem 13, we have that  (ℂ2,∥·∥ψ) is not n-uniformly convex for every n∈ℕ with n≥2. Therefore, we obtain the desired result.It is well known that a functionf from ℝ into ℝ defined by
(50)f(t)={0,ift≤0,exp⁡(-t-2),ift>0
for t∈ℝ is strictly increasing on [0,∞), infinitely differentiable and f(n)(0)=0 for all n∈ℕ.Example 15.
Defineγ∈Γ2 by
(51)γ(t)={-exp⁡(4-(t-12)-2),ift<12,0,ift=12,+exp⁡(4-(t-12)-2),ift>12
for t∈[0,1]. Then (ℂ2,∥·∥Sγ) is uniformly convex and not p-uniformly convex for all p∈[2,∞). See Figure 1.Figure 1Theorem 16.
Letγ∈Γ2 and q∈(1,2]. Assume that there exist a constant λ∈(0,1/2) and sequences {sn} and {un} in [0,1] such that sn<unfor n∈ℕ,
(52)lim⁡n→∞(un-sn)=0,lim⁡n→∞γ(λsn+(1-λ)un)-γ((1-λ)sn+λun)(un-sn)q-1=∞.
Then (ℂ2,∥·∥Sγ) is not q-uniformly smooth.Proof.
Putψ=Sγ. Without loss of generality, we may assume
(53)γ(λsn+(1-λ)un)-γ((1-λ)sn+λun)(un-sn)q-1≥n
for n∈ℕ, and {sn} and {un} converge to some number t0∈[0,1]. We define a sequence {tn} by (37). Since
(54)lim⁡n→∞ψ(sn)ψ(sn)+ψ(un)=12,lim⁡n→∞ψ(un)ψ(sn)+ψ(un)=12,
we may also assume that
(55)ψ(sn)ψ(sn)+ψ(un)∈[λ,1-λ],ψ(un)ψ(sn)+ψ(un)∈[λ,1-λ]
for n∈ℕ. We note that
(56)(1-λ)sn+λun≤tn≤λsn+(1-λ)un
because
(57)tn=ψ(un)ψ(sn)+ψ(un)sn+ψ(sn)ψ(sn)+ψ(un)un
for n∈ℕ. Define sequences {xn} and {yn} in ℂ2 by
(58)xn=1ψ(tn)(1-tn,tn),yn=(ψ(sn)(1-un)-ψ(un)(1-sn),ψ(sn)un-ψ(un)sn)(ψ(sn)+ψ(un))ψ(tn)
for n∈ℕ. It is obvious that ∥xn∥=1. We put vn∈[0,1] by (41). We have
(59)∥yn∥=∥(ψ(sn)(1-un)-ψ(un)(1-sn),ψ(sn)un-ψ(un)sn)∥(ψ(sn)+ψ(un))ψ(tn)=∥(ψ(sn)(un-1)+ψ(un)(1-sn),ψ(sn)un-ψ(un)sn)∥(ψ(sn)+ψ(un))ψ(tn)=ψ(sn)(2un-1)+ψ(un)(1-2sn)(ψ(sn)+ψ(un))ψ(tn)∥(1-vn,vn)∥=ψ(sn)(2un-1)+ψ(un)(1-2sn)(ψ(sn)+ψ(un))ψ(tn)ψ(vn)≤2(ψ(sn)(2un-1)+ψ(un)(1-2sn))≤4(un-sn)
by Lemma 7. We note that lim⁡n∥yn∥=0. We will calculate ∥xn+yn∥ and ∥xn-yn∥. We have
(60)xn+yn=1ψ(tn)(1-tn+ψ(sn)(1-un)-ψ(un)(1-sn)ψ(sn)+ψ(un),kxn+ynkkkkkkkktn+ψ(sn)un-ψ(un)snψ(sn)+ψ(un))xn+yn=1ψ(tn)(1+ψ(sn)-ψ(un)ψ(sn)+ψ(un))(1-un,un)xn+yn=1ψ(tn)2ψ(sn)ψ(sn)+ψ(un)(1-un,un)
because
(61)tn+((ψ(sn)un-ψ(un)sn)/(ψ(sn)+ψ(un)))1+((ψ(sn)-ψ(un))/(ψ(sn)+ψ(un)))=un.
Hence,
(62)∥xn+yn∥=1ψ(tn)2ψ(sn)ψ(un)ψ(sn)+ψ(un)
for n∈ℕ. Similarly, we obtain
(63)xn-yn=1ψ(tn)2ψ(un)ψ(sn)+ψ(un)(1-sn,sn)
and hence
(64)∥xn-yn∥=1ψ(tn)2ψ(sn)ψ(un)ψ(sn)+ψ(un)
for n∈ℕ. Therefore, we obtain
(65)ρ(∥yn∥)≥∥xn+yn∥+∥xn-yn∥2-1ρ(∥yn∥)=1ψ(tn)2ψ(sn)ψ(un)ψ(sn)+ψ(un)-1.
From
(66)2ψ(sn)ψ(un)-(ψ(un)+ψ(sn))ψ(tn)=ψ(sn)(ψ(un)-ψ(tn))-ψ(un)(ψ(tn)-ψ(sn))=ψ(sn)(∫tnλsn+(1-λ)un‍γ(s)ds+∫λsn+(1-λ)unun‍γ(s)ds)-ψ(un)(∫sn(1-λ)sn+λun‍γ(s)dskkkkkkkkkkkkkk+∫(1-λ)sn+λuntn‍γ(s)ds(∫sn(1-λ)sn+λun‍γ(s)ds)≥ψ(sn)γ(tn)(λsn+(1-λ)un-tn)+ψ(sn)γ(λsn+(1-λ)un)λ(un-sn)-ψ(un)γ((1-λ)sn+λun)λ(un-sn)-ψ(un)γ(tn)(tn-(1-λ)sn-λun)=ψ(sn)γ(tn)×(λsn+(1-λ)un-(sn/ψ(sn))+(un/ψ(un))(1/ψ(sn))+(1/ψ(un)))+ψ(sn)γ(λsn+(1-λ)un)λ(un-sn)-ψ(un)γ((1-λ)sn+λun)λ(un-sn)-ψ(un)γ(tn)×((sn/ψ(sn))+(un/ψ(un))(1/ψ(sn))+(1/ψ(un))-(1-λ)sn-λun)=-ψ(sn)γ(tn)λ(un-sn)+ψ(sn)γ(λsn+(1-λ)un)λ(un-sn)-ψ(un)γ((1-λ)sn+λun)λ(un-sn)+ψ(un)γ(tn)λ(un-sn)=ψ(sn)λ(un-sn)(γ(λsn+(1-λ)un)-γ(tn))+ψ(un)λ(un-sn)(γ(tn)-γ((1-λ)sn+λun))≥12λ(un-sn)(γ(λsn+(1-λ)un)-γ(tn))+12λ(un-sn)(γ(tn)-γ((1-λ)sn+λun))=12λ(un-sn)×(γ(λsn+(1-λ)un)-γ((1-λ)sn+λun))≥12λn(un-sn)q
and (59), we have
(67)ρ(∥yn∥)∥yn∥q≥1(4(un-sn))q(1/2)λn(un-sn)qψ(tn)(ψ(sn)+ψ(un))ρ(∥yn∥)∥yn∥q≥14q(1/2)λnψ(tn)(ψ(sn)+ψ(un))≥14q+1λn.
Hence we obtain limsup⁡τ→+0ρ(τ)/τq=∞. So by Proposition 2, we obtain the desired result.Corollary 17.
Letf be a bijective and strictly increasing function from [-1,1] into [0,1] with ∫-11‍f(a)da=1. Assume that f is infinitely differentiable on the neighborhood of some a0∈(-1,1), and
(68)f′(a0)=f′′(a0)=f′′′(a0)=⋯=0.
Then f-1∈Γ2 and (ℂ2,∥·∥Sf-1) is uniformly smooth and is not q-uniformly smooth for all q∈(1,2].Proof.
It is not difficult to checkf-1∈Γ2. Put γ=f-1 and ψ=Sγ. By Proposition 12, since γ is surjective, (ℂ2,∥·∥ψ) is smooth and hence it is uniformly smooth. Fix ν∈ℕ. As in the proof of Corollary 14, we can prove lim⁡b→a0+0(f(b)-f(a0))/(b-a0)ν=0. Since f is strictly increasing, we have
(69)lim⁡b→a0+0(b-a0)νf(b)-f(a0)=∞.
Putting u=f(b) and t0=f(a0), we have
(70)lim⁡u→t0+0γ(u)-γ(t0)(u-t0)1/ν=lim⁡b→a0+0b-a0(f(b)-f(a0))1/νlim⁡u→t0+0γ(u)-γ(t0)(u-t0)1/ν=lim⁡b→a0+0((b-a0)νf(b)-f(a0))1/νlim⁡u→t0+0γ(u)-γ(t0)(u-t0)1/ν=∞.
We choose a strictly increasing sequence {sn} and a strictly decreasing sequence {un} in [0,1] satisfying t0=(2/3)sn+(1/3)un for n∈ℕ and lim⁡nsn=lim⁡nun=t0. Then it is obvious that t0<(1/3)sn+(2/3)un for n∈ℕ and lim⁡n((1/3)sn+(2/3)un)=t0. We have
(71)lim⁡n→∞γ((1/3)sn+(2/3)un)-γ((2/3)sn+(1/3)un)(un-sn)1/ν=131/νlim⁡n→∞γ((1/3)sn+(2/3)un)-γ(t0)((1/3)sn+(2/3)un-t0)1/ν=∞.
Thus, by Theorem 16, we have that (ℂ2,∥·∥ψ) is not (1+1/ν)-uniformly smooth. Since ν is arbitrary, (ℂ2,∥·∥ψ) is not q-uniformly smooth for every q∈(1,2].Example 18.
Define a functionf from [-1,1] onto [0,1] by
(72)f(a)={-exp⁡(1-a-2)2+12,ifa<0,12,ifa=0,+exp⁡(1-a-2)2+12ifa>0
for a∈[-1,1]. Then (ℂ2,∥·∥Sf-1) is uniformly smooth and not q-uniformly smooth for all q∈(1,2]. See Figure 1.Example 19.
Letγ be as in Example 15 and let f be as in Example 18. Define a function η from [0,1] into [-1,1] by
(73)η(t)={f-1(4t)4-34,ift≤14,γ(2t-1/2)2,if14≤t≤34,f-1(4t-3)4+34,ift≥34
for t∈[0,1]. Then (ℂ2,∥·∥Sη) is uniformly convex, uniformly smooth, not p-uniformly convex for all p∈[2,∞), and not q-uniformly smooth for all q∈(1,2]. See Figure 1.



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*Source: 746309-2014-02-16.xml*