Homogeneity properties of subadditive functions

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1 Annaes Mathematicae et Informaticae pp Homogeneity properties of subadditive functions Pá Burai and Árpád Száz Institute of Mathematics, University of Debrecen e-mai: e-mai: Abstract We coect, suppement and extend some we-known basic facts on various homogeneity properties of subadditive functions. Key Words: Homogeneous and subadditive functions, seminorms and preseminorms. AMS Cassification Number: 39B72. Introduction Subadditive functions, with various homogeneity properties, pay important roes in many branches of mathematics. First of a, they occur in the Hahn- Banach theorems and the derivation of vector topoogies. See, for instance, [2] and [4]. A positivey homogeneous subadditive function is usuay caed subinear. Whie, an absoutey homogeneous subadditive function may be caed a seminorm. However, some important subadditive functions are ony preseminorms. Moreover, it is aso worth noticing that subbadditive functions are straightforward generaizations of the rea-vaued additive ones. Therefore, the study of additive functions shoud, in principe, be preceded by that of the subadditive ones. Subadditive functions have been intensivey studied by severa authors. Their most basic agebraic and anaytica properties have been estabished by R. Cooper [5], E. Hie [7, pp ], R. A. Rosenbaum [7], E. Berz [], M. Kuczma [9, pp ] and J. Matkowski []. In this paper, we are ony interested in the most simpe homogeneity properties of subadditive functions. Besides coecting some we-known basic facts, for instance, we prove the foowing theorem. 89

2 90 P. Burai and Á. Száz Theorem.. If p is a quasi-subadditive function of a vector space X over Q, and moreover x X and 0 k Z, then k p x p k x for a Z ; 2 p 3 p k x p k x p k x k p x for a 0 < Z ; k x k p x for a 0 > Z. Remark.2. If p is a subodd subadditive function of X, then p is additive. Therefore, the corresponding equaities are aso true. Whie, if p is an even subadditive function of X, then we can ony prove that for a x X, 0 k Z and Z. k p x p k x p k x 2. Superodd and subhomogeneous functions Definition 2.. A rea-vaued function p of a group X wi be caed subodd if p x p x for a x X; 2 superodd if p x p x for a x X. Remark 2.2. Note that thus p may be caed odd if it is both subodd and superodd. Moreover, p is superodd if and ony if p is subodd. Therefore, superodd functions need not be studied separatey. However, because of the forthcoming appications, it is more convenient to study superodd functions. By the above definition, we evidenty have the foowing Proposition 2.3. If p is a superodd function of a group X, then 0 p 0 ; 2 p x p x for a x X. Hint. Ceary, p 0 p 0 = p 0. Therefore, 0 2 p 0, and thus aso hods. Remark 2.4. Note that if p is subodd, then just the opposite inequaities hod. Therefore, if in particuar p is odd, then the corresponding equaities are aso true. Anaogousy to Definition 2., we may aso naturay introduce the foowing

3 Homogeneity properties of subadditive functions 9 Definition 2.5. A rea-vaued function p of a group X wi be caed N subhomogeneous if p n x n p x for a n N and x X; 2 N superhomogeneous if n p x p n x for a n N and x X. Remark 2.6. Note that thus p may be caed N homogeneous if it is both N subhomogeneous and N superhomogeneous. Moreover, p is N superhomogeneous if and ony if p is N subhomogeneous. Therefore, N superhomogeneous functions need not be studied separatey. Concerning N subhomogeneous functions, we can easiy estabish the foowing Proposition 2.7. If p is an N subhomogeneous function of a group X, then for a x X and 0 > k Z. p k x k p x. Proof. Under the above assumptions, we evidenty have p k x = p k x k p x = k p x. Now, as an immediate consequence of Definition 2.5 and Proposition 2.7, we can aso state Proposition 2.8. If p is an N subhomogeneous function of a group X x X, then k p k x p x for a 0 < k Z ; 2 p x k p k x for a 0 > k Z. Moreover, by using this proposition, we can easiy prove the foowing Theorem 2.9. If p is an N subhomogeneous function of a vector space X over Q, and moreover x X and Z, then k p x p k x for a 0 < k Z ; 2 k p x p k x for a 0 > k Z. and Proof. If 0 < k Z, then by Proposition 2.8 it is cear that k p x = k p k k x p k x. Whie, if 0 > k Z, then by the above inequaity it is cear that k p x = k p x p k x = p k x.

4 92 P. Burai and Á. Száz Remark 2.0. Note that if p is N superhomogeneous, then just the opposite inequaities hod. Therefore, if in particuar p is N homogeneous, then the corresponding equaities are aso true. 3. Subadditive and quasi-subadditive functions Foowing the terminoogy of Hie [7, p. 3 ] and Rosenbaum [7, p. 227 ], we may aso naturay have the foowing Definition 3.. A rea-vaued function p of a group X wi be caed subadditive if p x + y p x + p y for a x, y X; 2 superadditive if p x + p y p x + y for a x, y X. Definition 3.2. Note that thus p is additive if and ony if it is both subadditive and superadditive. Moreover, p is superadditive if and ony if p is subadditive. Therefore, superadditive functions need not be studied separatey. The appropriateness of Definitions 2. and 2.5 is apparent from the foowing theorem whose proof can aso be found in Kuczma [9, p. 40 ]. Theorem 3.3. If p is a subadditive function of a group X, then p is superodd and N subhomogeneous. Proof. Ceary, p 0 = p p 0 + p 0, and thus 0 p 0. Moreover, if x X, then we have 0 p 0 = p x + x p x + p x. Therefore, p x p x, and thus p is superodd. Moreover, if p n x n p x for some n N, then we aso have p n + x = p n x + x p n x + p x n p x + p x = n + p x. Hence, by the induction principe, it is cear that p n x n p x for a n N. Therefore, p is N subhomogeneous. Remark 3.4. Note that if p is superadditive, then p is subodd and N superhomogeneous. Therefore, if in particuar p is additive then p is odd and N homogeneous. Because of Theorem 3.3, we may aso naturay introduce the foowing

5 Homogeneity properties of subadditive functions 93 Definition 3.5. A rea-vaued function p of a group X wi be caed quasi-subadditive if it is superodd and N subhomogeneous ; 2 quasi-superadditive if it is subodd and N superhomogeneous. Remark 3.6. Note that thus p may be caed quasi-additive if it is both quasisubadditive and quasi-superadditive. Moreover, p is quasi-superadditive if and ony if p is quasi-subadditive. Therefore, quasi-superadditive functions need not be studied separatey. Now, in addition to Propositions 2.7 and 2.8, we can aso prove the foowing Proposition 3.7. If p is a quasi-subadditive function of a group X and x X, then k p x p k x for a 0 > k Z ; 2 k p k x p x for a 0 k Z. Proof. If 0 > k Z, then by the corresponding definitions we have Therefore, p k x p k x = p k x k p x = k p x. k p x p k x, and hence k p k x p x. Moreover, from Proposition 2.8 we know that the atter inequaity is aso true for 0 < k Z. Now, by using the above proposition, we can easiy prove the foowing counterpart of Theorem 2.9. Theorem 3.8. If p is a quasi subadditive function of a vector space X over Q, and moreover x X and 0 k Z, then k p x p k x for a Z ; 2 p 3 p k x p k x p k x k p x for a 0 < Z ; k x k p x for a 0 > Z. Proof. If Z, then by Proposition 3.7 2, it is cear that k p x = k p k k x p k x.

6 94 P. Burai and Á. Száz Moreover, if 0 < Z, then by using the N subhomogeneity of p and the = particuar case of Theorem 3.8 we can see that p k x = p k x p k x and k p x = k p x p k x. Whie, if 0 > Z, then by using Proposition 3.7 and the = particuar case of Theorem 3.8 we can see that p k x p k x = p k x and p k x k p x = k p x. Remark 3.9. Note that if p is quasi-superadditive, then just the opposite inequaities hod. Therefore, if p is in particuar quasi-additive, then the corresponding equaities are aso true. 4. Some further resuts on subadditive functions Whenever p is subadditive, then in addition to Theorem 3.8 we can aso prove the foowing Theorem 4.. If p is a subadditive function of a group X, then for any x, y X we have p x y p x p y p x y ; 2 p y + x p x p y p y + x. Proof. We evidenty have p x = p x y + y p x y + p y, and hence aso p y p y x + p x = p x y + p x. Therefore, is true. Moreover, quite simiary we aso have p x = p y y + x p y + p y + x, and hence aso p y p x + p x + y = p x + p y + x. Therefore, 2 is aso true.

7 Homogeneity properties of subadditive functions 95 Now, as a usefu consequence of the above theorem, we can aso state Coroary 4.2. If p is a subadditive function of a group X, then for any x, y X we have p x p y max { p x y, p x y } ; 2 p x p y max { p y + x, p y + x }. Proof. If then by Theorem 4. we have M = max { p x y, p x y }, p x p y p x y M and p x p y p x y M. Therefore, simiar. p x p y M, and thus is true. The proof of 2 is quite By using Theorem 4., we can easiy prove the foowing improvement of Kuczma s [9, Lemma 9, p. 402 ]. See aso Cooper [5, Theorem IX, p. 430 ]. Theorem 4.3. If p is a rea-vaued function of a group X, then the foowing assertions are equivaent : p is additive ; 2 p is odd and subadditive. 3 p is subodd and subadditive. Proof. If hods, then by Remark 3.4 it is cear that p is odd, and thus 2 aso hods. Therefore, since 2 triviay impies 3, we need actuay show that 3 impies. For this, note that if 3 hods, then by Remark 2.4 and Theorem 4. we have p x + p y p x + p y = p x p y p x y = p x + y for a x, y X. Hence, by the subadditivity of p, it is cear that aso hods. From the above theorem, by using Remark 3.9, we can immediatey get Coroary 4.4. If p is a subodd subadditive function of a vector space X over Q, then p r x = r p x for a r Q and x X. Hence, it is cear that in particuar we aso have Coroary 4.5. If p is a subodd subadditive function of Q, then p r = p r for a r Q.

8 96 P. Burai and Á. Száz 5. Even superodd and subhomogeneous functions Because of quasi-subadditive functions, it is aso worth studying even superodd and N subhomogeneous functions. Definition 5.. A rea-vaued function p of a group X wi be caed even if p x = p x for a x X. Remark 5.2. Now, in contrast to Definition 2., the subeven and supereven functions need not be introduced. Namey, we have evidenty the foowing Proposition 5.3. If p is a rea-vaued function of a group X, then the foowing assertions are equivaent : p is even ; 2 p x p x for a x X; 3 p x p x for a x X. Hint. If 3 hods, then for each x X we aso have p x p x = p x. Therefore, p x = p x, and thus aso hods. Remark 5.4. Note that a counterpart of the above proposition fais to hod for odd functions. Namey, if for instance p x = x for a x R, then p is superodd, but not odd. By using Definition 5., in addition to Proposition 2.3, we can aso easiy estabish the foowing extension of Cooper s [5, Theorem X, p. 430 ]. See aso Kuczma [9, Lemma 8, p. 402]. Proposition 5.5. If p is an even superodd function of a group X, then 0 p x for a x X. Proof. Namey, if x X, then p x p x = p x. Therefore, 0 2 p x, and thus 0 p x aso hods. Remark 5.6. Hence, it is cear that if p is an even subodd function of X, then p x 0 for a x X. Therefore, if in particuar p is an even and odd function of X, then we necessariy have p x = 0 for a x X. Moreover, by using Proposition 2.7 and Theorem 2.9, we can aso easiy prove the foowing counterparts of Proposition 3.7 and Theorem 3.8. Proposition 5.7. If p is an even N subhomogeneous function of a group X, then p k x k p x for a x X and 0 k Z.

9 Homogeneity properties of subadditive functions 97 Proof. If 0 < k Z, then the corresponding definitions we evidenty have p k x k p x = k p x. Whie, if 0 > k Z, then by Proposition 2.7 and the corresponding definitions we aso have p k x k p x = k p x. Theorem 5.8. If p is an even N subhomogeneous function of a vector space X over Q, then k p x p k x p k x for a x X and k, Z with k, 0. Proof. If k > 0, then Theorem 2.9 and Proposition 5.7 it is cear that k p x = k p x p k x = p k x p k x. Whie, if k < 0, then by Theorem and Proposition 5.7, it is cear that k p x = k p x p k x = p k x p k x. Remark 5.9. To compare the above theorem with Theorem 3.8, note that by the = particuar case of Theorem 5.8 now we aso have k p x p k x. Finay, we note that by Coroary 4.2 we can aso state the foowing Proposition 5.0. If p is an even subadditive function of a group X, then for a x, y X. p x p y min { p x y, p y + x }. 6. Homogeneous subadditive functions Definition 6.. A rea-vaued function p of a vector space X over K = R or C wi be caed homogeneous if p λ x = λ p x for a λ K and x X; 2 positivey homogeneous if p λ x = λ p x for a λ > 0 and x X; 3 absoutey homogeneous if p λ x = λ p x for a λ K and x X.

10 98 P. Burai and Á. Száz Remark 6.2. Note that if p is homogeneous absoutey homogeneous, then p is, in particuar, odd even and positivey homogeneous. Moreover, if p is positivey homogeneous, then in particuar we have p 0 = p 2 0 = 2 p 0, and hence p 0 = 0. Therefore, p 0 x = p 0 = 0 = 0 p x is aso true. Now, as some usefu characterizations of positivey and absoutey homogeneous functions, we can aso easiy prove the foowing two propositions. Proposition 6.3. If p is a rea-vaued function of a vector space X over R, then the foowing assertions are equivaent : p is positivey homogeneous ; 2 p λ x λ p x for a λ > 0 and x X ; 3 λ p x p λ x for a λ > 0 and x X. Proposition 6.4. If p is a rea-vaued function of a vector space X over K, then the foowing assertions are equivaent : p is absoutey homogeneous ; 2 p λ x λ p x for a 0 λ K and x X ; 3 λ p x p λ x for a 0 λ R and x X. Hint. If 3 hods, then for any 0 λ R and x X we aso have p λ x = λ p λ x λ p λ λ λ x = λ p x. Therefore, the corresponding equaity is aso true. Moreover, from Remark 6.2, we can see that p 0 x = p 0 = 0 = 0 p x. Therefore, aso hods. In addition to the above propositions, it is aso worth estabishing the foowing Theorem 6.5. If p is a rea-vaued function of a vector space X over R, then p is homogeneous if and ony if p is odd and positivey homogeneous ; 2 p is absoutey homogeneous if and ony if p is even and positivey homogeneous. Hint. If p is even and positivey homogeneous, then for any λ < 0 and x X we aso have p λ x = p λ x = λ p x = λ p x. Hence, by the second part of Remark 6.2, it is cear that p is absoutey homogeneous. Remark 6.6. >From Remark 6.2 and Theorem 4.3, we can see that a homogeneous subadditive function is necessariy inear. Therefore, ony some non-homogeneous subbadditive functions have to be studied separatey. The most important ones are the norms.

11 Homogeneity properties of subadditive functions 99 Definition 6.7. A rea-vaued, absoutey homogeneous, subadditive function p of a vector space X is caed a seminorm on X. In particuar, the seminorm p is caed a norm if p x 0 for a x X \{0}. Remark 6.8. Note that if p is a seminorm on X, then by Remark 6.2, Theorem 3.3 and Proposition 5.5, we necessariy have 0 p x for a x X. Definition 6.9. A rea-vaued subadditive function p of a vector space X over K is caed a preseminorm on X if im λ 0 p λ x = 0 for a x X; 2 p λ x p x for a x X and λ K with λ. In particuar, the preseminorm p is caed a prenorm if p x 0 for a x X \ {0}. Remark 6.0. By Remark 6.8, it is cear that every seminorm p on X is, in particuar, a preseminorm. Moreover, if p is a preseminorm on X, then defining p x = min {, p x } or p x = p x/ + p x for a x X, it can be shown that p is a preseminorm on X such that p is not a seminorm. Most of the foowing basic properties of preseminoms have aso been estabished in [8]. The simpe proofs are incuded here for the reader s convenience. Theorem 6.. If p is a preseminorm on a vector space X over K and x X, then p 0 = p x ; 3 p λ x = p λ x for a λ K ; 4 p x p y p x y for a y X; 5 p λ x p µ x for a λ, µ K with λ µ ; 6 p λ x n p x for a λ K and n N with λ n ; 7 k p x p k x p k x for a k, Z with k 0. Proof. By Definition 6.9, we evidenty have p 0 = im λ 0 p 0 = im λ 0 p λ 0 = 0. Moreover, if λ, µ K such that λ µ and µ 0, then by using Definition we can see that p λ x = p λ/µ µ x p µ x. Hence, since λ µ and µ = 0 impy λ = 0, it is cear 5 is aso true.

12 200 P. Burai and Á. Száz Now, by 5 and the inequaities λ λ λ, it is cear that in particuar we aso have p λ x p λ x p λ x. Therefore, 3 is aso true. Moreover, if n N such that λ n, then by 5 and Theorem 3.3, it is cear that p λ x p n x n p x aso hods. Finay, to compete the proof, we note that by 3 p is, in particuar, even. Therefore, by Propositions 5.3 and 5.0 and Theorem 5.8, assertions 2, 4 and 7 are aso true. Remark 6.2. From the above proof, it is cear that if p is a subadditive function of a vector space X over K such that in addition to Definition we ony have inf λ 0 p λ x 0 for a x X, then p is aready a preseminorm. Finay, we note that by using Theorem 6. 2 and 6 we can aso prove Coroary 6.3. If p is a nonzero preseminorm on a one-dimensiona vector space X over K, then p is necessariy a prenorm on X. Proof. Namey, if this not the case, then there exists x X such that x 0 and p x = 0. Hence, by using dim X =, we can see that X = K x. Moreover, if λ K, then by choosing n N such that λ n we can see that 0 p λ x n p x = 0, and thus p λ x = 0. Therefore, p is identicay zero, which is a contradiction. Remark 6.4. The importance of preseminorms ies mainy in the fact that in contrast to seminorms, a nonzero preseminorm can be bounded by Remark 6.0. Thus, by an idea of Fréchet, any sequence p n n= preseminorms on X can be repaced by a singe preseminorm q = n= /2n pn which induces the same topoogy on X. In this respect, it is aso worth mentioning that, in contrast to seminorms, each vector topoogy on X can be derived from a famiy preseminorms on X. See, for instance, [4]. References [] Berz, E., Subinear functions on R, Aequationes Math., Vo , [2] Buskes, G., The Hahn Banach theorem surveyed, Dissertationes Math., Vo. 327, [3] Bruckner, A. M., Minima superadditive extensions of superadditive functions, Pacific J. Math., Vo , [4] Bruckner, A. M., Some reationships between ocay superadditive functions and convex functions, Proc. Amer. Math. Soc., Vo , 6 65.

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