HERMITE-HADAMARD-TYPE INEQUALITIES FOR GENERALIZED CONVEX FUNCTIONS MIHÁLY BESSENYEI Institute of Mthemtics University of Debrecen H-4010 Debrecen, Pf. 12, Hungry EMil: besse@mth.klte.hu Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Received: 21 June, 2008 Accepted: 14 July, 2008 Communicted by: P.S. Bullen 2000 AMS Sub. Clss.: Primry 26A51, 26B25, 26D15. Key words: Abstrct: Hermite Hdmrd inequlity, generlized convexity, Beckenbch fmilies, Chebyshev systems, Mrkov Krein theory. The im of the present pper is to extend the clssicl Hermite-Hdmrd inequlity to the cse when the convexity notion is induced by Chebyshev system. Pge 1 of 101 Go Bck Acknowledgement: This reserch hs been supported by the Hungrin Scientific Reserch Fund (OTKA) Grnts NK 68040.
1 Introduction 3 2 Polynomil Convexity 7 2.1 Orthogonl polynomils nd bsic qudrture formule....... 8 2.2 An pproximtion theorem...................... 18 2.3 Hermite Hdmrd-type inequlities................. 22 2.4 Applictions.............................. 33 3 Generlized 2-Convexity 38 3.1 Chrcteriztions vi generlized lines................ 38 3.2 Connection with stndrd convexity................. 52 3.3 Hermite Hdmrd-type inequlities................. 56 3.4 Applictions.............................. 59 4 Generlized Convexity Induced by Chebyshev Systems 63 4.1 Chrcteriztions nd regulrity properties.............. 64 4.2 Moment spces induced by Chebyshev systems........... 67 4.3 Hermite Hdmrd-type inequlities................. 72 4.4 An lterntive pproch in prticulr cse............. 78 5 Chrcteriztions vi Hermite Hdmrd Inequlities 86 5.1 Further properties of generlized lines................ 86 5.2 Hermite Hdmrd-type inequlities nd (ω 1, ω 2 )-convexity.... 91 Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 2 of 101 Go Bck
1. Introduction Let I be rel intervl, tht is, nonempty, connected nd bounded subset of R. An n-dimensionl Chebyshev system on I consists of set of rel vlued continuous functions ω 1,..., ω n nd is determined by the property tht ech n points of I R with distinct first coordintes cn uniquely be interpolted by liner combintion of the functions. More precisely, we hve the following Definition 1.1. Let I R be rel intervl nd ω 1,..., ω n : I R be continuous functions. Denote the column vector whose components re ω 1,..., ω n in turn by ω, tht is, ω := (ω 1,..., ω n ). We sy tht ω is Chebyshev system over I if, for ll elements x 1 < < x n of I, the following inequlity holds: ω(x 1 ) ω(x n ) > 0. In fct, it suffices to ssume tht the determinnt bove is nonvnishing whenever the rguments x 1,..., x n re pirwise distinct points of the domin. Indeed, Bolzno s theorem gurntees tht its sign is constnt if the rguments re supposed to be in n incresing order, hence the components ω 1,..., ω n cn lwys be rerrnged such tht ω fulfills the requirement of the definition. However, considering Chebyshev systems s vectors of functions insted of sets of functions is widely ccepted in the technicl literture nd lso turns out to be very convenient in our investigtions. Without climing completeness, let us list some importnt nd clssicl exmples of Chebyshev systems. In ech exmple ω is defined on n rbitrry I R except for the lst one where I ] π 2, π 2 [. polynomil system: ω(x) := (1, x,..., x n ); exponentil system: ω(x) := (1, exp x,..., exp nx); hyperbolic system: ω(x) := (1, cosh x, sinh x,..., cosh nx, sinh nx); Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 3 of 101 Go Bck
trigonometric system: ω(x) := (1, cos x, sin x,..., cos nx, sin nx). We mke no ttempt here to present n exhustive ccount of the theory of Chebyshev systems, but only mention tht, motivted by some results of A.A. Mrkov, the first systemtic investigtion of the geometric theory of Chebyshev systems ws done by M. G. Krein. However, let us note tht Chebyshev systems ply n importnt role, sometimes indirectly, in numerous fields of mthemtics, for exmple, in the theory of pproximtion, numericl nlysis nd the theory of inequlities. The books [16] nd [15] contin rich literture nd bibliogrphy of the topics for the interested reder. The notion of convexity cn lso be extended by pplying Chebyshev systems: Definition 1.2. Let ω = (ω 1,..., ω n ) be Chebyshev system over the rel intervl I. A function f : I R is sid to be generlized convex with respect to ω if, for ll elements x 0 < < x n of I, it stisfies the inequlity ( 1) n f(x 0 ) f(x n ) ω(x 0 ) ω(x n ) 0. There re other lterntives to express tht f is generlized convex with respect to ω, for exmple, f is generlized ω-convex or simply ω-convex. If the underlying n-dimensionl Chebyshev system cn uniquely be identified from the context, we briefly sy tht f is generlized n-convex. If ω is the polynomil Chebyshev system, the definition leds to the notion of higher-order monotonicity which ws introduced nd studied by T. Popoviciu in sequence of ppers [20, 22, 21, 24, 23, 27, 29, 25, 30, 28, 26, 31, 33, 32, 34, 35]. A summry of these results cn be found in [36] nd [17]. For the ske of uniform terminology, throughout the this pper Popoviciu s setting is clled polynomil convexity. Tht is, function f : I R is sid to be polynomilly n-convex if, for ll Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 4 of 101 Go Bck
elements x 0 < < x n of I, it stisfies the inequlity f(x 0 )... f(x n ) 1... 1 ( 1) n x 0... x n. 0..... x n 1 0... x n 1 n Observe tht polynomilly 2-convex functions re exctly the stndrd convex ones. The cse, when the generlized convexity notion is induced by the specil two dimensionl Chebyshev system ω 1 (x) := 1 nd ω 2 (x) := x, is termed stndrd setting nd stndrd convexity, respectively. The integrl verge of ny stndrd convex function f : [, b] R cn be estimted from the midpoint nd the endpoints of the domin s follows: ( ) + b f 2 1 b f(x)dx f() + f(b). 2 This is the well known Hdmrd s inequlity ([11]) or, s it is quoted for historicl resons (see [12], [18] for interesting remrks), the Hermite-Hdmrd-inequlity. The im of this pper is to verify nlogous inequlities for generlized convex functions, tht is, to give lower nd upper estimtions for the integrl verge of the function using certin bse points of the domin. Of course, the bse points re supposed to depend only on the underlying Chebyshev system of the induced convexity. For this purpose, we shll follow n inductive pproch since it seems to hve more dvntges thn the deductive one. First of ll, it mkes the originl motivtions cler; on the other hnd, it llows us to use the most suitble mthemticl tools. Hence sophisticted proofs tht sometimes occur when using deductive pproch cn lso be voided. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 5 of 101 Go Bck
SECTION 2 investigtes the cse of polynomil convexity. The bse points of the Hermite Hdmrd-type inequlities turn out to be the zeros of certin orthogonl polynomils. The min tools of the section re bsed on some methods of numericl nlysis, like the Guss qudrture formul nd Hermite-interpoltion. A smoothing technique nd two theorems of Popoviciu re lso crucil. In SECTION 3 we present Hermite Hdmrd-type inequlities for generlized 2-convex functions. The most importnt uxiliry result of the proof is chrcteriztion theorem which, in the stndrd setting, reduces to the well known chrcteriztion properties of convex functions. Another theorem of the section estblishes tight reltionship between stndrd nd generlized 2-convexity. This result hs importnt regulrity consequences nd is lso essentil in verifying Hermite Hdmrd-type inequlities. The generl cse is studied in SECTION 4. The min results gurntee only the existence nd lso the uniqueness of the bse points of the Hermite Hdmrd-type inequlities but offer no explicit formule for determining them. The min tool of the section is the Krein Mrkov theory of moment spces induced by Chebyshev systems. In some specil cses (when the dimension of the underlying Chebyshev systems re smll ), n elementry lterntive pproch is lso presented. SECTION 5 is devoted to showing tht, t lest in the two dimensionl cse nd requiring wek regulrity conditions, Hermite Hdmrd-type inequlities re not merely the consequences of generlized convexity, but they lso chrcterize it. Specilizing the members of Chebyshev systems, severl pplictions nd exmples re presented for concrete Hermite Hdmrd-type inequlities in both the cses of polynomil convexity nd generlized 2-convexity. As simple consequence, the clssicl Hermite Hdmrd inequlity is mong the corollries in ech cse s well. The results of this pper cn be found in [3, 4, 5, 6, 7] nd [1]. In wht follows, we present them without ny further references to the mentioned ppers. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 6 of 101 Go Bck
2. Polynomil Convexity The min results of this section stte Hermite Hdmrd-type inequlities for polynomilly convex functions. Let us recll tht function f : I R is sid to be polynomilly n-convex if, for ll elements x 0 < < x n of I, it stisfies the inequlity f(x 0 )... f(x n ) 1... 1 ( 1) n x 0... x n. 0..... x n 1 0... x n 1 n In order to determine the bse points nd coefficients of the inequlities, Guss-type qudrture formule re pplied. Then, using the reminder term of the Hermiteinterpoltion, the min results follow immeditely for sufficiently smooth functions due to the next two theorems of Popoviciu: Theorem A. ([17, Theorem 1. p. 387]) Assume tht f : I R is continuous nd n times differentible on the interior of I. Then, f is polynomilly n-convex if nd only if f (n) 0 on the interior of I. Theorem B. ([17, Theorem 1. p. 391]) Assume tht f : I R is polynomilly n-convex nd n 2. Then, f is (n 2) times differentible nd f (n 2) is continuous on the interior of I. To drop the regulrity ssumptions, smoothing technique is developed tht gurntees the pproximtion of polynomilly convex functions with smooth polynomilly convex ones. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 7 of 101 Go Bck
2.1. Orthogonl polynomils nd bsic qudrture formule In wht follows, ρ denotes positive, loclly integrble function (briefly: weight function) on n intervl I. The polynomils P nd Q re sid to be orthogonl on [, b] I with respect to the weight function ρ or simply ρ-orthogonl on [, b] if P, Q ρ := P Qρ = 0. A system of polynomils is clled ρ-orthogonl polynomil system on [, b] I if ech member of the system is ρ-orthogonl to the others on [, b]. Define the moments of ρ by the formule µ k := x k ρ(x)dx (k = 0, 1, 2,...). Then, the n th degree member of the ρ-orthogonl polynomil system on [, b] hs the following representtion vi the moments of ρ: 1 µ 0 µ n 1 x µ P n (x) := 1 µ n....... x n µ n µ 2n 1 Clerly, it suffices to show tht P n is ρ-orthogonl to the specil polynomils 1, x,..., x n 1. Indeed, for k = 1,..., n, the first nd the (k + 1) st columns of the determinnt P n (x), x k 1 ρ re linerly dependent ccording to the definition of the moments. In fct, the moments nd the orthogonl polynomils depend hevily on the intervl [, b]. Therefore, we use the notions µ k;[,b] nd P n;[,b] insted of µ k nd P n bove when we wnt to or hve to emphsize the dependence on the underlying intervl. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 8 of 101 Go Bck
Throughout this section, the following property of the zeros of orthogonl polynomils plys key role (see [39]). Let P n denote the n th degree member of the ρ-orthogonl polynomil system on [, b]. Then, P n hs n pirwise distinct zeros ξ 1 < < ξ n in ], b[. Let us consider the following (2.1) (2.2) (2.3) (2.4) fρ = n c k f(ξ k ), fρ = c 0 f() + fρ = n c k f(ξ k ), n c k f(ξ k ) + c n+1 f(b), fρ = c 0 f() + n c k f(ξ k ) + c n+1 f(b). Guss-type qudrture formule where the coefficients nd the bse points re to be determined so tht (2.1), (2.2), (2.3) nd (2.4) re exct when f is polynomil of degree t most 2n 1, 2n, 2n nd 2n+1, respectively. The subsequent four theorems investigte these cses. Theorem 2.1. Let P n be the n th degree member of the orthogonl polynomil system on [, b] with respect to the weight function ρ. Then (2.1) is exct for polynomils f of degree t most 2n 1 if nd only if ξ 1,..., ξ n re the zeros of P n, nd Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 9 of 101 Go Bck (2.5) c k = P n (x) (x ξ k )P n(ξ k ) ρ(x)dx.
Furthermore, ξ 1,..., ξ n re pirwise distinct elements of ], b[, nd c k 0 for ll k = 1,..., n. This theorem follows esily from well known results in numericl nlysis (see [13], [14], [39]). For the ske of completeness, we provide proof. Proof. First ssume tht ξ 1,..., ξ n re the zeros of the polynomil P n nd, for ll k = 1,..., n, denote the primitive Lgrnge-interpoltion polynomils by L k : [, b] R. Tht is, L k (x) := P n (x) (x ξ k )P n(ξ k ) if x ξ k 1 if x = ξ k. If Q is polynomil of degree t most 2n 1, then, using the Euclidin lgorithm, Q cn be written in the form Q = P P n + R where deg P, deg R n 1. The inequlity deg P n 1 implies the ρ-orthogonlity of P nd P n : P P n ρ = 0. On the other hnd, deg R n 1 yields tht R is equl to its Lgrnge-interpoltion polynomil: n R = R(ξ k )L k. Therefore, considering the definition of the coefficients c 1,..., c n in formul (2.5), Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 10 of 101 Go Bck
we obtin tht Qρ = = P P n ρ + n c k R(ξ k ) = Rρ = n R(ξ k ) L k ρ n ( c k P (ξk )P n (ξ k ) + R(ξ k ) ) = n c k Q(ξ k ). Tht is, the qudrture formul (2.1) is exct for polynomils of degree t most 2n 1. Conversely, ssume tht (2.1) is exct for polynomils of degree t most 2n 1. Define the polynomil Q by the formul Q(x) := (x ξ 1 ) (x ξ n ) nd let P be polynomil of degree t most n 1. Then, deg P Q 2n 1, nd thus P Qρ = c 1 P (ξ 1 )Q(ξ 1 ) + + c n P (ξ n )Q(ξ n ) = 0. Therefore Q is ρ-orthogonl to P. The uniqueness of P n implies tht P n = n Q, nd ξ 1,..., ξ n re the zeros of P n. Furthermore, (2.1) is exct if we substitute f := L k nd f := L 2 k, respectively. The first substitution gives (2.5), while the second one shows the nonnegtivity of c k. For further detils, consult the book [39, p. 44]. Theorem 2.2. Let P n be the n th degree member of the orthogonl polynomil system on [, b] with respect to the weight function ρ (x) := (x )ρ(x). Then (2.2) is exct for polynomils f of degree t most 2n if nd only if ξ 1,..., ξ n re the zeros of P n, nd c 0 = 1 (2.6) P P n(x)ρ(x)dx, 2 n() 2 (2.7) c k = 1 ξ k (x )P n (x) (x ξ k )P n(ξ k ) ρ(x)dx. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 11 of 101 Go Bck
Furthermore, ξ 1,..., ξ n re pirwise distinct elements of ], b[, nd c k 0 for ll k = 0,..., n. Proof. Assume tht the qudrture formul (2.2) is exct for polynomils of degree t most 2n. If P is polynomil of degree t most 2n 1, then P ρ = (x )P (x)ρ(x)dx = c 1 (ξ 1 )P (ξ 1 ) + + c n (ξ n )P (ξ n ). Applying Theorem 2.1 to the weight function ρ nd the coefficients c ;k := c k (ξ k ), we get tht ξ 1,..., ξ n re the zeros of P n nd, for ll k = 1,..., n, the coefficients c ;k cn be computed using formul (2.5). Therefore, c k (ξ k ) = P n (x) (x ξ k )P n(ξ k ) ρ (x)dx = Substituting f := P 2 n into (2.1), we obtin tht c 0 = 1 P 2 n() P 2 nρ. (x )P n (x) (x ξ k )P n(ξ k ) ρ(x)dx. Thus (2.6) nd (2.7) re vlid, nd c k 0 for k = 0, 1,..., n. Conversely, ssume tht ξ 1,..., ξ n re the zeros of the orthogonl polynomil P n, nd the coefficients c 1,..., c n re given by the formul (2.7). Define the coefficient c 0 by c 0 = ρ (c 1 + + c n ). If P is polynomil of degree t most 2n, then there exists polynomil Q with deg Q 2n 1 such tht Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 12 of 101 Go Bck P (x) = (x )Q(x) + P ().
Indeed, the polynomil P (x) P () vnishes t the point x =, hence it is divisible by (x ). Applying Theorem 2.1 gin to the weight function ρ, Qρ = c ;1 Q(ξ 1 ) + + c ;n Q(ξ n ) holds. Thus, using the definition of c 0, the representtion of the polynomil P nd the qudrture formul bove, we hve tht P (x)ρ(x)dx = = ( (x )Q(x) + P () ) ρ(x)dx n c k (ξ k )Q(ξ k ) + = c 0 P () + = c 0 P () + n P ()c k k=0 n ( c k (ξk )Q(ξ k ) + P () ) n c k P (ξ k ), which yields tht the qudrture formul (2.2) is exct for polynomils of degree t most 2n. Therefore, substituting f := P 2 n into (2.2), we get formul (2.6). Theorem 2.3. Let P n be the n th degree member of the orthogonl polynomil system on [, b] with respect to the weight function ρ b (x) := (b x)ρ(x). Then (2.3) is exct for polynomils f of degree t most 2n if nd only if ξ 1,..., ξ n re the zeros of P n, Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 13 of 101 Go Bck
nd (2.8) (2.9) c k = 1 b ξ k c n+1 = 1 Pn(b) 2 (b x)p n (x) (x ξ k )P n(ξ k ) ρ(x)dx, P 2 n(x)ρ(x)dx. Furthermore, ξ 1,..., ξ n re pirwise distinct elements of ], b[, nd c k 0 for ll k = 1,..., n + 1. Hint. Applying similr rgument to the previous one to the weight function ρ b, we obtin the sttement of the theorem. Theorem 2.4. Let P n be the n th degree member of the orthogonl polynomil system on [, b] with respect to the weight function ρ b. Then (2.4) is exct for polynomils f of degree t most 2n + 1 if nd only if ξ 1,..., ξ n re the zeros of P n, nd (2.10) (2.11) (2.12) c 0 = c k = c n+1 = 1 (b )P 2 n() 1 (b ξ k )(ξ k ) 1 (b )P 2 n(b) (b x)p 2 n(x)ρ(x)dx, (b x)(x )P n (x) ρ(x)dx, (x ξ k )P n(ξ k ) (x )P 2 n(x)ρ(x)dx. Furthermore, ξ 1,..., ξ n re pirwise distinct elements of ], b[, nd c k 0 for ll k = 0,..., n + 1. Proof. Assume tht the qudrture formul (2.4) is exct for polynomils of degree Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 14 of 101 Go Bck
t most 2n + 1. If P is polynomil of degree t most 2n 1, then P ρ b = (b x)(x )P (x)ρ(x)dx = c 1 (b ξ 1 )(ξ 1 )P (ξ 1 ) + + c n (b ξ n )(ξ n )P (ξ n ). Applying Theorem 2.1 to the weight function ρ b nd the coefficients c,b;k := c k (b ξ k )(ξ k ), we get tht ξ 1,..., ξ n re the zeros of P n nd, for ll k = 1,..., n, the coefficients c,b;k cn be computed using formul (2.5). Therefore, c k (b ξ k )(ξ k ) = = P n (x) (x ξ k )P n(ξ k ) ρb (x)dx (b x)(x )P n (x) ρ(x)dx. (x ξ k )P n(ξ k ) Substituting f := (b x)p 2 n(x) nd f := (x )P 2 n(x) into (2.1), we obtin tht c 0 = c n+1 = 1 (b )Pn() 2 1 (b )P 2 n(b) (b x)p 2 n(x)ρ(x)dx, (x )P 2 n(x)ρ(x)dx. Thus (2.10), (2.11) nd (2.12) re vlid, furthermore, c k 0 for k = 0,..., n + 1. Conversely, ssume tht ξ 1,..., ξ n re the zeros of P n, nd the coefficients c 1,..., c n Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 15 of 101 Go Bck
re given by the formul (2.11). Define the coefficients c 0 nd c n+1 by the equtions n (b x)ρ(x)dx = c 0 (b ) + c k (b ξ k ), (x )ρ(x)dx = n c k (ξ k ) + c n+1 (b ). If P is polynomil of degree t most 2n + 1, then there exists polynomil Q with deg Q 2n 1 such tht (b )P (x) = (b x)(x )Q(x) + (x )P (b) + (b x)p (). Indeed, the polynomil (b )P (x) (x )P (b) (b x)p () is divisible by (b x)(x ) since x = nd x = b re its zeros. Applying Theorem 2.1 gin, Qρ b = c,b;1 Q(ξ 1 ) + + c,b;n Q(ξ n ) holds. Thus, using the definition of c 0 nd c n+1, the representtion of the polynomil P nd the qudrture formul bove, we hve tht (b ) = = P (x)ρ(x)dx ( (b x)(x )Q(x) + (x )P (b) + (b x)p () ) ρ(x)dx n c k (b ξ k )(ξ k )Q(ξ k ) + P (b) (x )ρ(x)dx + P () (b x)ρ(x)dx Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 16 of 101 Go Bck
= = n c k (b ξ k )(ξ k )Q(ξ k ) + c 0 (b )P () + + n c k (b ξ k )P () n c k (ξ k )P (b) + c n+1 (b )P (b) n ( c k (b ξk )(ξ k )Q(ξ k ) + (ξ k )P (b) + (b ξ k )P () ) + c 0 (b )P () + c n+1 (b )P (b) n = c 0 (b )P () + c k (b )P (ξ k ) + c n+1 (b )P (b), which yields tht the qudrture formul (2.4) is exct for polynomils of degree t most 2n + 1. Therefore, substituting f := (b x)p 2 n(x) nd f := (x )P 2 n(x) into (2.4), formule (2.10) nd (2.12) follow. Let f : [, b] R be differentible function, x 1,..., x n be pirwise distinct elements of [, b], nd 1 r n be fixed integer. We denote the Hermite interpoltion polynomil by H, which stisfies the following conditions: H(x k ) = f(x k ) H (x k ) = f (x k ) (k = 1,..., n), (k = 1,..., r). Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 17 of 101 Go Bck We recll tht deg H = n + r 1. From well known result, (see [13, Sec. 5.3, pp.
230-231]), for ll x [, b] there exists θ such tht (2.13) f(x) H(x) = ω n(x)ω r (x) f (n+r) (θ), (n + r)! where 2.2. An pproximtion theorem ω k (x) = (x x 1 ) (x x k ). It is well known tht there exists function ϕ which possesses the following properties: (i) ϕ : R R + is C, i. e., it is infinitely mny times differentible; (ii) supp ϕ [ 1, 1]; (iii) R ϕ = 1. Using ϕ, one cn define the function ϕ ε for ll ε > 0 by the formul ϕ ε (x) = 1 ( x ) ε ϕ (x R). ε Then, s it cn esily be checked, ϕ ε stisfies the following conditions: (i ) ϕ ε : R R + is C ; Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 18 of 101 Go Bck (ii ) supp ϕ ε [ ε, ε]; (iii ) R ϕ ε = 1.
Let I R be nonempty open intervl, f : I R be continuous function, nd choose ε > 0. Denote the convolution of f nd ϕ ε by f ε, tht is f ε (x) := f(y)ϕ ε (x y)dy (x R) R where f(y) = f(y) if y I, otherwise f(y) = 0. Let us recll, tht f ε f uniformly s ε 0 on ech compct subintervl of I, nd f ε is infinitely mny times differentible on R. These importnt results cn be found for exmple in [40, p. 549]. Theorem 2.5. Let I R be n open intervl, f : I R be polynomilly n- convex continuous function. Then, for ll compct subintervls [, b] I, there exists sequence of polynomilly n-convex nd C functions (f k ) which converges uniformly to f on [, b]. Proof. Choose, b I nd ε 0 > 0 such tht the inclusion [ ε 0, b + ε 0 ] I holds. We show tht the function τ ε f : [, b] R defined by the formul τ ε f(x) := f(x ε) is polynomilly n-convex on [, b] for 0 < ε < ε 0. Let x 0 < < x n b nd k n 1 be fixed. By induction, we re going to verify the identity τ ε f(x 0 ) τ ε f(x n ) τ ε f(x 0 ) τ ε f(x n ) 1 1 1 1 x 0 x n x 0 ε x n ε.......... (2.14) x k 1 0 x k 1 = n (x 0 ε) k 1 (x n ε) k 1. x k 0 x k n x k 0 x k n.......... x0 n 1 x n 1 n x n 1 0 x n 1 n Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 19 of 101 Go Bck
If k = 1, then this eqution obviously holds. Assume, for fixed positive integer k n 2, tht (2.14) remins true. The binomil theorem implies the identity x k = ( ) k ε k + 0 ( ) k ε k 1 (x ε) + + 1 ( ) k (x ε) k. k Tht is, (x ε) k is the liner combintion of the elements 1, x ε,..., (x ε) k nd x k. Therefore, dding the pproprite liner combintion of the 2 nd,..., (k + 1) st rows to the (k + 2) nd row, we rrive t the eqution τ ε f(x 0 ) τ ε f(x n ) 1 1 x 0 ε x n ε..... (x 0 ε) k 1 (x n ε) k 1 = x k 0 x k n x k+1 0 x k+1 n..... x n 1 0 x n 1 n τ ε f(x 0 ) τ ε f(x n ) 1 1 x 0 ε x n ε..... (x 0 ε) k 1 (x n ε) k 1. (x 0 ε) k (x n ε) k x k+1 0 x k+1 n..... x n 1 0 x n 1 n Hence formul (2.14) holds for ll fixed positive k whenever 1 k n 1. The prticulr cse k = n 1 gives the polynomil n-convexity of τ ε f. Applying chnge of vribles nd the previous result, we get tht Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 20 of 101 Go Bck
f ε (x 0 ) f ε (x n ) 1 1 ( 1) n x 0 x n..... x n 1 0 x n 1 n f(t)ϕ ε (x 0 t) f(t)ϕε (x n t) 1 1 = ( 1) n x 0 x n dt R..... x n 1 0 x n 1 n f(x 0 s) f(xn s) 1 1 = ( 1) n x 0 x n R. ϕ ε (s)ds.... x n 1 0 x n 1 n τ s f(x 0 ) τ s f(x n ) 1 1 = ( 1) n x 0 x n R. ϕ ε (s)ds 0,.... x n 1 0 x n 1 n which shows the polynomil n-convexity of f ε on [, b] for 0 < ε < ε 0. To complete the proof, choose positive integer n 0 such tht the reltion 1 n 0 < ε 0 holds. If we define ε k nd f k by ε k := 1 n 0 +k nd f k := f εk for k N, then 0 < ε k < Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 21 of 101 Go Bck
ε 0, nd thus (f k ) stisfies the requirements of the theorem. 2.3. Hermite Hdmrd-type inequlities In the sequel, we shll need two dditionl uxiliry results. The first one investigtes the convergence properties of the zeros of orthogonl polynomils. Lemm 2.1. Let ρ be weight function on [, b], nd ( j ) be strictly monotone decresing, (b j ) be strictly monotone incresing sequences such tht j, b j b nd 1 < b 1. Denote the zeros of P m;j by ξ 1;j,..., ξ m;j, where P m;j is the m th degree member of the ρ [j,b j ]-orthogonl polynomil system on [ j, b j ], nd denote the zeros of P m by ξ 1,..., ξ m, where P m is the m th degree member of the ρ-orthogonl polynomil system on [, b]. Then, lim j ξ k;j = ξ k (k = 1,..., n). Proof. Observe first tht the mpping (, b) µ k;[,b] is continuous, therefore µ k;[j,b j ] µ k;[,b] hence P m;j P m pointwise ccording to the representtion of orthogonl polynomils. Tke ε > 0 such tht ]ξ k ε, ξ k + ε[ ], b[, ]ξ k ε, ξ k + ε[ ]ξ l ε, ξ l + ε[= (k l, k, l {1,..., m}). The polynomil P m chnges its sign on ]ξ k ε, ξ k + ε[ since it is of degree m nd it hs m pirwise distinct zeros; therefore, due to the pointwise convergence, P m;j lso chnges its sign on the sme intervl up to n index. Tht is, for sufficiently lrge j, ξ k;j ]ξ k ε, ξ k + ε[. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 22 of 101 Go Bck The other uxiliry result investigtes the one-sided limits of polynomilly n- convex functions t the endpoints of the domin. Let us note tht its first ssertion involves, in fct, two cses ccording to the prity of the convexity.
Lemm 2.2. Let f : [, b] R be polynomilly n-convex function. Then, (i) ( 1) n f() lim sup t +0 ( 1) n f(t); (ii) f(b) lim sup t b 0 f(t). Proof. It suffices to restrict the investigtions to the even cse of ssertion (i) only since the proofs of the other ones re completely the sme. For the ske of brevity, we shll use the nottion f + () := lim sup t +0 f(t). Tke the elements x 0 := < x 1 := t < < x n of [, b]. Then, the (even order) polynomil convexity of f implies f() f(t) f(x 2 )... f(x n ) 1 1 1... 1 t x 2... x n 0........ n 1 t n 1 x n 1 2... x n 1 n Therefore, tking the limsup s t + 0, we obtin tht f() f + () f(x 2 )... f(x n ) 1 1 1... 1 x 2... x n....... n 1 n 1 x n 1 2... x n 1 n 0. The djoint determinnts of the elements f(x 2 ),..., f(x n ) in the first row re equl to zero since their first nd second columns coincide; on the other hnd, f() nd f + () hve the sme (positive) Vndermonde-type djoint determinnt. Hence, pplying the expnsion theorem on the first row, we obtin the desired inequlity f() f + () 0. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 23 of 101 Go Bck
The min results concern the cses of odd nd even order polynomil convexity seprtely in the subsequent two theorems. Theorem 2.6. Let ρ : [, b] R be positive integrble function. Denote the zeros of P m by ξ 1,..., ξ m where P m is the m th degree member of the orthogonl polynomil system on [, b] with respect to the weight function (x )ρ(x), nd denote the zeros of Q m by η 1,..., η m where Q m is the m th degree member of the orthogonl polynomil system on [, b] with respect to the weight function (b x)ρ(x). Define the coefficients α 0,..., α m nd β 1,..., β m+1 by the formule nd α 0 := 1 P 2 m() α k := 1 ξ k P 2 m(x)ρ(x)dx, (x )P m (x) (x ξ k )P m(ξ k ) ρ(x)dx β k := 1 (b x)q m (x) b η k (x η k )Q m(η k ) ρ(x)dx, β m+1 := 1 Q 2 Q m(x)ρ(x)dx. 2 m(b) If function f : [, b] R is polynomilly (2m + 1)-convex, then it stisfies the following Hermite Hdmrd-type inequlity m m α 0 f() + α k f(ξ k ) fρ β k f(η k ) + β m+1 f(b). Proof. First ssume tht f is (2m + 1) times differentible. Then, ccording to Theorem A, f (2m+1) 0 on ], b[. Let H be the Hermite interpoltion polynomil Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 24 of 101 Go Bck
determined by the conditions H() = f(), H(ξ k ) = f(ξ k ), H (ξ k ) = f (ξ k ). By the reminder term (2.13) of the Hermite interpoltion, if x is n rbitrry element of ], b[, then there exists θ ], b[ such tht f(x) H(x) = (x )(x ξ 1) 2 (x ξ m ) 2 f (2m+1) (θ). (2m + 1)! Tht is, fρ Hρ on [, b] due to the nonnegtivity of f (2m+1) nd the positivity of ρ. On the other hnd, H is of degree 2m, therefore Theorem 2.2 yields tht fρ Hρ = α 0 H() + m α k H(ξ k ) = α 0 f() + m α k f(ξ k ). For the generl cse, let f be n rbitrry polynomilly (2m+1)-convex function. Without loss of generlity we my ssume tht m 1; in this cse, f is continuous (see Theorem B). Let ( j ) nd (b j ) be sequences fulfilling the requirements of Lemm 2.1. According to Theorem 2.5, there exists sequence of C, polynomilly (2m + 1)-convex functions (f i;j ) such tht f i;j f uniformly on [ j, b j ] s i. Denote the zeros of P m;j by ξ 1;j,..., ξ m;j where P m;j is the m th degree member of the orthogonl polynomil system on [ j, b j ] with respect to the weight function (x )ρ(x). Define the coefficients α 0;j,..., α m;j nlogously to α 0,..., α m with the help of P m;j. Then, ξ k;j ξ k due to Lemm 2.1, nd hence α k;j α k s j. Applying the previous step of the proof on the smooth functions (f i;j ), it Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 25 of 101 Go Bck
follows tht α 0;j f i;j ( j ) + m α k;j f i;j (ξ k;j ) j j f i;j ρ. Tking the limits i nd then j, we get the inequlity ( ) α 0 lim inf f(t) + t +0 m α k f(ξ k ) This, together with Lemm 2.2, gives the left hnd side inequlity to be proved. The proof of the right hnd side inequlity is nlogous, therefore it is omitted. The second min result offers Hermite Hdmrd-type inequlities for evenorder polynomilly convex functions. In this cse, the symmetricl structure disppers: the lower estimtion involves none of the endpoints, while the upper estimtion involves both of them. Theorem 2.7. Let ρ : [, b] R be positive integrble function. Denote the zeros of P m by ξ 1,..., ξ m where P m is the m th degree member of the orthogonl polynomil system on [, b] with respect to the weight function ρ(x), nd denote the zeros of Q m 1 by η 1,..., η m 1 where Q m 1 is the (m 1) st degree member of the orthogonl polynomil system on [, b] with respect to the weight function (b x)(x )ρ(x). Define the coefficients α 1,..., α m nd β 0,..., β m+1 by the formule P m (x) α k := (x ξ k )P m(ξ k ) ρ(x)dx fρ. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 26 of 101 Go Bck
nd β 0 = β k = β m+1 = 1 (b )Q 2 m 1() 1 (b η k )(ξ k ) 1 (b )Q 2 m 1(b) (b x)q 2 m 1(x)ρ(x)dx, (b x)(x )Q m 1 (x) ρ(x)dx, (x η k )Q m 1(η k ) (x )Q 2 m 1(x)ρ(x)dx. If function f : [, b] R is polynomilly (2m)-convex, then it stisfies the following Hermite Hdmrd-type inequlity m m 1 α k f(ξ k ) fρ β 0 f() + β k f(η k ) + β m f(b). Proof. First ssume tht f is n = 2m times differentible. Then f (2m) 0 on ], b[ ccording to Theorem B. Consider the Hermite interpoltion polynomil H tht interpoltes the function f in the zeros of P m in the following mnner: H(ξ k ) = f(ξ k ), H (ξ k ) = f (ξ k ). By the reminder term (2.13) of the Hermite interpoltion, if x is n rbitrry element of ], b[, then there exists θ ], b[ such tht f(x) H(x) = (x ξ 1) 2 (x ξ m ) 2 f (2m) (θ). (2m)! Hence fρ Hρ on [, b] due to the nonnegtivity of f (2m) nd the positivity of ρ. On the other hnd, H is of degree 2m 1, therefore Theorem 2.1 yields the left hnd Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 27 of 101 Go Bck
side of the inequlity to be proved: fρ Hρ = m α k H(ξ k ) = m α k f(ξ k ). Now consider the Hermite interpoltion polynomil H tht interpoltes the function f t the zeros of Q m 1 nd t the endpoints of the domin in the following wy: H() = f(), H(η k ) = f(η k ), H (η k ) = f (η k ), H(b) = f(b). By the reminder term (2.13) of the Hermite interpoltion, if x is n rbitrry element of ], b[, then there exists θ ], b[ such tht f(x) H(x) = (x )(x b)(x η 1) 2 (x η m 1 ) 2 f (2m) (θ). (2m)! The fctors of the right hnd side re nonnegtive except for the fctor (x b) which is negtive, hence fρ Hρ. On the other hnd, H is of degree 2m 1, therefore Theorem 2.4 yields the right hnd side inequlity to be proved: fρ Hρ = β 0 H() + m 1 m 1 β k H(η k ) + β m H(b) = β 0 f() + β k f(η k ) + β m f(b). Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 28 of 101 Go Bck
From this point, n nlogous rgument to the corresponding prt of the previous proof gives the sttement of the theorem without ny differentibility ssumptions on the function f. Specilizing the weight function ρ 1, the roots of the inequlities cn be obtined s convex combintions of the endpoints of the domin. The coefficients of the convex combintions re the zeros of certin orthogonl polynomils on [0, 1] in both cses. Observe tht interchnging the role of the endpoints in ny side of the inequlity concerning the odd order cse, we obtin the other side of the inequlity. Theorem 2.8. Let, for m 0, the polynomil P m be defined by the formul 1 1 1 2 m+1 1 1 x 3 m+2 P m (x) :=....... x m 1 1 m+2 2m+1 Then, P m hs m pirwise distinct zeros λ 1,..., λ m in ]0, 1[. Define the coefficients α 0,..., α m by α 0 := 1 P 2 m(0) α k := 1 λ k 1 0 1 0 P 2 m(x)dx, xp m (x) (x λ k )P m(λ k ) dx. If function f : [, b] R is polynomilly (2m + 1)-convex, then it stisfies the Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 29 of 101 Go Bck
following Hermite Hdmrd-type inequlity α 0 f() + m α k f ( (1 λ k ) + λ k b ) 1 f(x)dx b m α k f ( λ k + (1 λ k )b ) + α 0 f(b). Proof. Apply Theorem 2.6 in the prticulr setting when := 0, b := 1 nd the weight function is ρ 1. Then, s simple clcultions show, P m is exctly the m th degree member of the orthogonl polynomil system on [0, 1] with respect to the weight function ρ(x) = x (see the beginning of this section). Therefore, P m hs m pirwise distinct zeros 0 < λ 1 < < λ m < 1. Moreover, the coefficients α 0,..., α m hve the form bove. Define the function F : [0, 1] R by the formul F (t) := f ( (1 t) + tb ). It is esy to check tht F is polynomilly (2m + 1)-convex on the intervl [0, 1]. Hence, pplying Theorem 2.6 nd the previous observtions, it follows tht 1 0 F (t)dt α 0 F (0) + = α 0 f() + m α k F (λ k ) m α k f ( (1 λ k ) + λ k b ). On the other hnd, to complete the proof of the left hnd side inequlity, observe tht Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 30 of 101 Go Bck 1 b f(x)dx = 1 0 F (t)dt.
For verifying the right hnd side one, define the function ϕ : [, b] R by the formul ϕ(x) := f( + b x). Then, ϕ is polynomilly (2m + 1)-convex on [, b]. The previous inequlity pplied on ϕ gives the upper estimtion of the Hermite Hdmrd-type inequlity for f. Theorem 2.9. Let, for m 1, the polynomils P m nd Q m 1 be defined by the formule 1 1 1 m 1 1 x 2 m+1 P m (x) :=.....,. x m 1 1 m+1 2m 1 1 1 2 3 m(m+1) 1 1 x 3 4 (m+1)(m+2) Q m 1 (x) :=....... x m 1 1 1 (m+1)(m+2) (2m 1)2m Then, P m hs m pirwise distinct zeros λ 1,..., λ m in ]0, 1[ nd Q m 1 hs m 1 pirwise distinct zeros µ 1,..., µ m 1 in ]0, 1[, respectively. Define the coefficients α 1,..., α m nd β 0,..., β m by α k := 1 0 P m (x) (x λ k )P m(λ k ) dx Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 31 of 101 Go Bck
nd β 0 := 1 Q 2 m 1(0) 1 1 β k := (1 µ k )µ k 1 β m := Q 2 m 1(1) (1 x)q 2 m 1(x)dx, 0 1 0 1 0 x(1 x)q m 1 (x) (x µ k )Q m 1(µ k ) dx, xq 2 m 1(x)dx. If function f : [, b] R is polynomilly (2m)-convex, then it stisfies the following Hermite Hdmrd-type inequlity m α k f ( (1 λ k ) + λ k b ) 1 f(x)dx b β 0 f() + m 1 β k f ( (1 µ k ) + µ k b ) + β m f(b). Proof. Substitute := 0, b := 1 nd ρ 1 into Theorem 2.7. Then, P m is exctly the m th degree member of the orthogonl polynomil system on the intervl [0, 1] with respect to the weight function ρ(x) = 1; similrly, Q m 1 is the (m 1) st degree member of the orthogonl polynomil system on the intervl [0, 1] with respect to the weight function ρ(x) = (1 x)x. Therefore, Q m hs m pirwise distinct zeros 0 < λ 1 < < λ m < 1 nd Q m 1 hs m 1 pirwise distinct zeros 0 < µ 1 < < µ m 1 < 1. Moreover, the coefficients α 1,..., α m nd β 0,..., β m hve the form bove. To complete the proof, pply Theorem 2.7 on the function F : [0, 1] R defined by the formul F (t) := f ( (1 t) + tb ). Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 32 of 101 Go Bck
2.4. Applictions In the prticulr setting when m = 1, Theorem 2.8 reduces to the clssicl Hermite Hdmrd inequlity: Corollry 2.1. If f : [, b] R is polynomilly 2-convex (i.e. convex) function, then the following inequlities hold ( ) + b f 1 f() + f(b) f(x)dx. 2 b 2 In the subsequent corollries we present Hermite Hdmrd-type inequlities in those cses when the zeros of the polynomils in Theorem 2.8 nd Theorem 2.9 cn explicitly be computed. Corollry 2.2. If f : [, b] R is polynomilly 3-convex function, then the following inequlities hold 1 4 f() + 3 ( ) + 2b 4 f 1 f(x)dx 3 ( ) 2 + b 3 b 4 f + 1 3 4 f(b). Corollry 2.3. If f : [, b] R is polynomilly 4-convex function, then the following inequlities hold ( 1 2 f 3 + 3 6 + 3 ) ( 3 b + 12 6 f 1 b 3 3 6 + 3 + ) 3 b 6 f(x)dx 1 6 f() + 2 3 f ( + b 2 ) + 1 6 f(b). Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 33 of 101 Go Bck
Corollry 2.4. If f : [, b] R is polynomilly 5-convex function, then the following inequlities hold ( 1 16 + 6 4 + 6 f() + f + 6 ) 6 b 9 36 10 10 + 16 ( 6 4 6 f + 6 + ) 6 b 36 10 10 1 b 16 6 f 36 f(x)dx ( 6 + 6 10 ( + 16 + 6 f 36 + 4 ) 6 b 10 6 6 10 + 4 + 6 b 10 ) + 1 9 f(b). In some other cses nlogous sttements cn be formulted pplying Theorem 2.9. For simplicity, insted of writing down these corollries explicitly, we shll present list which contins the zeros of P n (denoted by λ k ), the coefficients α k for the left hnd side inequlity, lso the zeros of Q n (denoted by µ k ), nd the coefficients β k for the right hnd side inequlity, respectively. Cse n = 6 The zeros of P 3 : 5 15 1, 10 2, 5 + 15 ; 10 the corresponding coefficients: 5 18, 4 9, 5 18. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 34 of 101 Go Bck
The zeros of Q 2 : the corresponding coefficients: Cse n = 8 The zeros of P 4 : 5 5, 10 1 525 + 70 30 2, 70 1 525 70 30 2 +, 70 the corresponding coefficients: 1 4 30 72, 1 4 + The zeros of Q 3 : 1 2 the corresponding coefficients: 5 + 5 ; 10 1 12, 5 12, 5 12, 1 12. 1 525 70 30 2, 70 1 525 + 70 30 2 + ; 70 30 72, 1 30 4 + 72, 1 30 4 72. 21 14, 1 2, 1 21 2 + 14 ; 1 20, 49 180, 16 45, 49 180, 1 20. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 35 of 101 Go Bck
Cse n = 10 The zeros of P 5 : 1 245 + 14 70 2, 42 1 245 14 70 2, 42 1 2, 1 245 14 70 2 + 1 245 + 14 70, 42 2 + ; 42 the corresponding coefficients: 322 13 70 322 + 13 70 64,, 1800 1800 225, 322 + 13 70, 1800 The zeros of Q 4 : 1 147 + 42 7 2, 42 1 147 42 7 2 +, 42 the corresponding coefficients: 1 30, 14 7, 60 14 + 7, 60 Cse n = 12 (right hnd side inequlity) The zeros of Q 5 : 1 147 42 7 2, 42 1 147 + 42 7 2 + ; 42 14 + 7, 60 14 7, 60 322 13 70. 1800 1 30. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 36 of 101 Go Bck 1 495 + 66 15 2, 66 1 495 66 15 2, 66
1 2, 1 495 66 15 2 +, 66 the corresponding coefficients: 1 495 + 66 15 2 + ; 66 1 42, 124 7 15, 700 124 + 7 15, 700 128 525, 124 + 7 15 124 7 15 1,, 700 700 42. During the investigtions of the higher order cses bove, we cn use the symmetry of the zeros of the orthogonl polynomils with respect to 1/2, nd therefore the clcultions led to solving liner or qudrtic equtions. The first cse where csus irreducibilis ppers is n = 7; similrly, this is the reson for presenting only the right hnd side inequlity for polynomilly 12-convex functions. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 37 of 101 Go Bck
3. Generlized 2-Convexity In terms of geometry, the Chebyshev property of two dimensionl system cn equivlently be formulted: the liner combintions of the members of the system (briefly: generlized lines) re continuous; furthermore, ny two points of the plin with distinct first coordintes cn be connected by unique generlized line. Tht is, generlized lines hve the most importnt properties of ffine functions. These properties turn out to be so strong tht most of the clssicl results of stndrd convexity, cn be generlized for this setting. First we investigte some bsic properties of generlized lines of two dimensionl Chebyshev systems. Then the most importnt tool of the section, chrcteriztion theorem is proved for generlized 2-convex functions. Two consequences of this theorem, nmely the existence of generlized support lines nd the property of generlized chords re crucil to verify Hermite Hdmrd-type inequlities. Another result sttes tight connection between stndrd nd (ω 1, ω 2 )-convexity, nd lso gurntees the integrbility of (ω 1, ω 2 )-convex functions. Some clssicl results of the theory of convex functions, like their representtion nd stbility re lso generlized for this setting. 3.1. Chrcteriztions vi generlized lines Let us recll tht (ω 1, ω 2 ) is sid to be Chebyshev system over n intervl I if ω 1, ω 2 : I R re continuous functions nd, for ll elements x < y of I, ω 1(x) ω 1 (y) ω 2 (x) ω 2 (y) > 0. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 38 of 101 Go Bck Some concrete exmples on Chebyshev systems re presented in the lst section of the section. Given Chebyshev system (ω 1, ω 2 ), function f : I R is clled
generlized convex with respect to (ω 1, ω 2 ) or briefly: generlized 2-convex if, for ll elements x < y < z of I, it stisfies the inequlity f(x) f(y) f(z) ω 1 (x) ω 1 (y) ω 1 (z) ω 2 (x) ω 2 (y) ω 2 (z) 0. Clerly, in the stndrd setting this definition reduces to the notion of (ordinry) convexity. Let (ω 1, ω 2 ) be Chebyshev system on n intervl I, nd denote the set of ll liner combintions of the functions ω 1 nd ω 2 by (ω 1, ω 2 ). We sy tht function ω : I R is generlized line if it belongs to the liner hull (ω 1, ω 2 ). The properties of generlized lines ply the key role in our further investigtions; first we need the following simple but useful ones. Lemm 3.1. Let (ω 1, ω 2 ) be Chebyshev system over n intervl I. Then, two different generlized lines of (ω 1, ω 2 ) hve t most one common point; moreover, if two different generlized lines hve the sme vlue t some ξ I, then the difference of the lines is positive on one side of ξ while negtive on the other side of ξ. In prticulr, ω 1 nd ω 2 hve t most one zero; moreover, if ω 1 (resp., ω 2 ) vnishes t some ξ I, then ω 1 is positive on one side of ξ while negtive on the other. Proof. Due to the liner structure of (ω 1, ω 2 ), without loss of generlity we my ssume tht one of the lines is the constnt zero line. Then, the other generlized line ω hs the representtion αω 1 + βω 2, with α 2 + β 2 > 0. The first ssertion of the theorem is equivlent to the property tht ω hs t most one zero. To show this, ssume indirectly tht ω(ξ) nd ω(η) equl zero for ξ η; tht is, αω 1 (ξ) + βω 2 (ξ) = 0, αω 1 (η) + βω 2 (η) = 0. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 39 of 101 Go Bck
By the Chebyshev property of (ω 1, ω 2 ), the bse determinnt of the system is nonvnishing, therefore the system hs only trivil solutions α = 0 nd β = 0 which contrdicts the property α 2 + β 2 > 0. An equivlent formultion of the second ssertion is the following: if ω(ξ) = 0 for some interior point ξ, then ω > 0 on one side of ξ while ω < 0 on the other. If this is not true, then, ccording to the previous result nd Bolzno s theorem, ω is strictly positive (or negtive) on both sides of ξ. For simplicity, ssume tht ω(t) > 0 for t ξ. Define the generlized line ω by ω := βω 1 + αω 2. Then, (ω, ω ) is lso Chebyshev system: if x < y re elements of I, then ω(x) ω(ξ) ω (x) ω (y) = α β β α ω 1(x) ω 1 (y) ω 2 (x) ω 2 (y) = (α 2 + β 2 ) ω 1(x) ω 1 (y) ω 2 (x) ω 2 (y) > 0. Therefore, tking the elements x < ξ < y of I, we rrive t the inequlities 0 < ω(x) ω(ξ) ω (x) ω (ξ) = ω(x)ω (ξ), 0 < ω(ξ) ω(y) ω (ξ) ω (y) = ω(y)ω (ξ), which yields the contrdiction tht ω (ξ) is simultneously positive nd negtive. For the lst ssertion, notice tht ω 1, ω 2 nd the constnt zero functions re specil generlized lines nd pply the previous prt of the theorem. The most importnt property of (ω 1, ω 2 ) gurntees the existence of generlized line prllel to the constnt zero function, which itself is generlized line s well (see below). Moreover, s it cn lso be shown, (ω 1, ω 2 ) fulfills the xioms of hyperbolic geometry. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 40 of 101 Go Bck
Lemm 3.2. If (ω 1, ω 2 ) is Chebyshev system on n intervl I, then there exists ω (ω 1, ω 2 ) such tht ω is positive on I. Proof. If ω 1 hs no zero in I, then ω := ω 1 or ω := ω 1 (ccording to the sign of ω 1 ) will do. Suppose tht ω 1 (ξ) = 0 for some ξ I. Due to Lemm 3.1, without loss of generlity we my ssume tht ω 1 (x) < 0 ω 1 (y) > 0 (x < ξ, x I), (y > ξ, y I). Hermite-Hdmrd-type Inequlities Choose the elements x < ξ < y of I. The Chebyshev property of (ω 1, ω 2 ) nd the negtivity of ω 1 (x)ω 2 (y) implies the inequlity Hence (3.1) α := sup y>ξ ω 2 (y) ω 1 (y) < ω 2(x) ω 1 (x). [ ] ω2 (y) inf ω 1 (y) x<ξ [ ] ω2 (x) ; ω 1 (x) moreover, both sides re rel numbers. We show tht the generlized line defined by ω := αω 1 ω 2 is positive on the interior of I. First observe tht ω tkes positive vlue t the point ξ. Indeed, by the definition of ω we hve ω(ξ) := αω 1 (ξ) ω 2 (ξ) = ω 2 (ξ); on the other hnd, for y > ξ, the positivity of ω 1 (y) nd the Chebyshev property of (ω 1, ω 2 ) yields ω 2 (ξ) > 0. If y > ξ, then the definition of α implies vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 41 of 101 Go Bck α ω 2(y) ω 1 (y) ;
multiplying both sides by the positive ω 1 (y) nd rerrnging the terms we get, ω(y) := αω 1 (y) ω 2 (y) 0. If x < ξ, then inequlity (3.1) gives tht α ω 2(x) ω 1 (x) ; multiplying both sides by the negtive ω 1 (x) nd rerrnging the obtined terms, we rrive t the inequlity ω(x) := αω 1 (x) ω 2 (x) 0. To complete the proof, it suffices to show tht ω lwys differs from zero on the interior of the domin. Assume indirectly tht ω(η) := αω 1 (η) ω 2 (η) = 0 for some η I. Clerly, η ξ since ω(ξ) > 0. Therefore, ω 1 (η) 0 nd α cn be expressed explicitly: α = ω 2(η) ω 1 (η). If ξ < η, choose y I such tht η < y hold. By the positivity of ω 1 (η)ω 1 (y) nd the Chebyshev property of (ω 1, ω 2 ), α = ω 2(η) ω 1 (η) < ω 2(y) ω 1 (y) which contrdicts the definition of α. Similrly, if ξ > η, choose x I such tht x < η is vlid. Then, the positivity of ω 1 (x)ω 1 (η) nd the Chebyshev property of (ω 1, ω 2 ) imply the inequlity Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 42 of 101 Go Bck α = ω 2(η) ω 1 (η) > ω 2(x) ω 1 (x), which contrdicts (3.1).
As n importnt consequence of Lemm 3.2, Chebyshev system cn lwys be replced equivlently by regulr one. In other words, ssuming positivity on the first component of Chebyshev system, s is required in mny further results, is not n essentil restriction. Moreover, the next lemm lso gives chrcteriztion of pirs of functions to form Chebyshev system. Lemm 3.3. Let (ω 1, ω 2 ) be Chebyshev system on n intervl I R. Then, there exists Chebyshev system (ω 1, ω 2) on I tht possesses the following properties: (i) ω 1 is positive on I ; (ii) ω 2/ω 1 is strictly monotone incresing on I ; (iii) (ω 1, ω 2 )-convexity is equivlent to (ω 1, ω 2)-convexity. Conversely, if ω 1, ω 2 : I R re continuous functions such tht ω 1 is positive nd ω 2 /ω 1 is strictly monotone incresing, then (ω 1, ω 2 ) is Chebyshev system over I. Proof. Lemm 3.2 gurntees the existence of rel constnts α nd β such tht αω 1 + βω 2 > 0 holds for ll x I. Define the functions ω 1, ω 2 : I R by the formule ω 1 := αω 1 + βω 2, ω 2 := βω 1 + αω 2. Choosing the elements x < y of I nd pplying the product rule of determinnts, we get ω 1(x) ω1(y) ω2(x) ω2(y) = α β β α ω 1(x) ω 1 (y) ω 2 (x) ω 2 (y) = (α 2 + β 2 ) ω 1(x) ω 1 (y) ω 2 (x) ω 2 (y) > 0. Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 43 of 101 Go Bck
Therefore, (ω1, ω2) is lso Chebyshev system over I. Assuming tht ω1 is positive, s it cn esily be checked, the Chebyshev property of (ω1, ω2) yields tht the function ω2/ω 1 is strictly monotone incresing on the interior of I. Lstly, let f : I R be n rbitrry function nd x < y < z be rbitrry elements of I. Then, by the product rule of determinnts, f(x) f(y) f(z) ω1(x) ω1(y) ω1(z) ω2(x) ω2(y) ω2(z) = 1 0 0 0 α β 0 β α = (α 2 + β 2 ) f(x) f(y) f(z) ω 1 (x) ω 1 (y) ω 1 (z) ω 2 (x) ω 2 (y) ω 2 (z) f(x) f(y) f(z) ω 1 (x) ω 1 (y) ω 1 (z) ω 2 (x) ω 2 (y) ω 2 (z) which shows tht the function f is generlized convex with respect to the Chebyshev system (ω 1, ω 2 ) if nd only if it is generlized convex with respect to the Chebyshev system (ω 1, ω 2). The proof of the converse ssertion is simple clcultion, therefore it is omitted. The following result gives vrious chrcteriztions of (ω 1, ω 2 )-convexity vi the monotonicity of the generlized divided difference, the generlized support property nd the locl nd the globl generlized chord properties. Theorem 3.1. Let (ω 1, ω 2 ) be Chebyshev system over n intervl I such tht ω 1 is positive on I. The following sttements re equivlent: (i) f : I R is (ω 1, ω 2 )-convex;, Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 Title Pge Pge 44 of 101 Go Bck
(ii) for ll elements x < y < z of I we hve tht f(y) f(z) ω 1 (y) ω 1 (z) f(x) ω 1 (x) ω 1(y) ω 1 (z) ω 2 (y) ω 2 (z) ω 1(x) ω 2 (x) f(y) ω 1 (y) ω 1 (y) ω 2 (y) ; (iii) for ll x 0 I there exist α, β R such tht αω 1 (x 0 ) + βω 2 (x 0 ) = f(x 0 ), αω 1 (x) + βω 2 (x) f(x) (x I); Hermite-Hdmrd-type Inequlities vol. 9, iss. 3, rt. 63, 2008 (iv) for ll n N, x 0, x 1,..., x n I nd λ 1,..., λ n 0 stisfying the conditions n λ k ω 1 (x k ) = ω 1 (x 0 ), we hve tht n λ k ω 2 (x k ) = ω 2 (x 0 ), f(x 0 ) n λ k f(x k ); (v) for ll x 0, x 1, x 2 I nd λ 1, λ 2 0 stisfying the conditions we hve tht λ 1 ω 1 (x 1 ) + λ 2 ω 1 (x 2 ) = ω 1 (x 0 ), λ 1 ω 2 (x 1 ) + λ 2 ω 2 (x 2 ) = ω 2 (x 0 ), f(x 0 ) λ 1 f(x 1 ) + λ 2 f(x 2 ); Title Pge Pge 45 of 101 Go Bck