Aequt. Mth. 91 017, 49 444 c The Authors 017. This rticle is published with open ccess t Springerlink.com 0001-9054/17/03049-16 published online Mrch 10, 017 DOI 10.1007/s00010-017-047-0 Aequtiones Mthemtice On the K-Riemnn integrl nd Hermite Hdmrd inequlities for K-convex functions Andrzej Olbryś Abstrct. In the present pper we introduce notion of the K-Riemnn integrl s nturl generliztion of usul Riemnn integrl nd study its properties. The im of this pper is to extend the clssicl Hermite Hdmrd inequlities to the cse when the usul Riemnn integrl is replced by the K-Riemnn integrl nd the convexity notion is replced by K-convexity. Mthemtics Subject Clssifiction. 6A51, 6B5, 6D15. Keywords. K-convexity, K-Riemnn integrl, Rdil K-derivtive, Hermite Hdmrd inequlities. 1. Introduction Throughout this pper I R stnds for n intervl nd K denotes subfield of the field of rel numbers R. Clerly, Q K, where Q denotes the field of rtionl numbers. We denote the set of the positive elements of K by K +. In the sequel the symbol [, b] A will denote n A-convex hull of the set {, b}, where A R i.e. [, b] A = {α +1 αb : α A [0, 1]}. In the cse when A = R we will use the stndrd symbol [, b] insted of [, b] R. Definition 1. A mpping f : R R is clled dditive if it stisfies Cuchy s functionl eqution fx + y =fx+fy, for every x, y R. A mpping f is clled K-liner if f is dditive nd K- homogeneous i.e. fαx =αfx,
430 Andrzej Olbryś AEM is fulfilled for every x R nd α K. It is well-known tht every dditive function is Q-homogeneous. Definition. A function f : I R is sid to be Jensen-convex if x + y f fx+fy, for every x, y I. A mp f is clled K-convex if fαx +1 αy αfx+1 αfy, for every x, y I nd α K 0, 1. It is known tht given function f is Jensen-convex if nd only if it is Q- convex see [,9]. On the other hnd, if f is K-convex then it is lso Q-convex. In this plce we introduce the following definitions Definition 3. A function f : I R is clled rdilly K-continuous t point x 0 I if for every u I lim f1 αx 0 + αu =fx 0. K + α 0 We sy tht f is rdilly K-continuous if it is rdilly K-continuous t every point from the domin. Definition 4. We sy tht function f : I R is uniformly rdilly K- continuous if for ny x 0 I nd u I the mpping [0, 1] K α fx 0 + αu x 0 is uniformly continuous. It is esy to see tht ny continuous nd ny uniformly continuous function f : I R in the usul sense is rdilly K-continuous, nd uniformly rdilly K-continuous, respectively. However, it cn hppen tht uniformly rdilly K-continuous function is discontinuous t every point in the usul sense. An esy exmple is provided by ny discontinuous K-liner mp. On the other hnd, every uniformly rdilly K-continuous function is lso rdilly K-continuous, but the converse is not true. We strt with the following esy-to-prove propositions. Proposition 5. Afunctionf : I R is rdilly K-continuous if nd only if for every, b I the function f [,b]k is continuous. Proposition 6. Afunctionf : I R is uniformly rdilly K-continuous if nd only if for ny, b I the mp f [,b]k is uniformly continuous.
Vol. 91 017 On K-Riemnn integrl nd the Hermite Hdmrd 431. Construction of the K-Riemnn integrl Now, we introduce notion of the K-Riemnn integrl s nturl generliztion of the clssicl Riemnn integrl. For the theory of the clssicl Riemnn integrl see for instnce [10, 14, 15]. Let P [,b] denote the set of prtitions of the intervl [, b] i.e. P [,b] := {t 0,t 1,...,t n : = t 0 <t 1 < <t n = b}. n=1 Following Zs. Páles [1] we define the set of K-prtitions of the intervl [, b] in the following wy { P[,b] K := t 0,t 1,...,t n P [,b] : t i } b K, i=1,,...,n { = t 0,t 1,...,t n P [,b] : t i = + α i b :α i 1 } K [0, 1],i=1,,...,n { } = t 0,t 1,...,t n P [,b] : t i [, b] K,i=1,,...,n. Now, suppose tht f :[, b] R is bounded function on the set [, b] K with M := sup fx, m := inf fx. x [,b] K x [,b] K For given K-prtition π =t 0,t 1,...,t n P[,b] K let M i := sup fx, m i := inf fx, i =1,,...,n. x [t i 1,t i] K x [t i 1,t i] K These suprem nd infim re well-defined, finite rel numbers since f is bounded on [, b] K.Moreover, m m i M i M, i =1,,...,n. We define the upper K-Riemnn sum of f with respect to the prtition π by n U K f,π := M i t i t i 1, nd the lower K-Riemnn sum of f with respect to the prtition π by n L K f,π := m i t i t i 1. Note tht i=1 i=1 mb L K f,π U K f,π Mb.
43 Andrzej Olbryś AEM Now, we define the upper K-Riemnn integrl of f on [, b] by { } ftd K t := inf U K f,π : π P[,b] K nd the lower K-Riemnn integrl by { } ftd K t := sup L K f,π : π P[,b] K. Definition 7. A function f :[, b] R bounded on [, b] K is sid to be K- Riemnn integrble on [, b] if its upper nd lower integrls re equl. In tht cse, the K-Riemnn integrl of f on [, b] is denoted by ftd K t. In the cse when K = R we will use the stndrd symbol ftdt insted of ftd Rt. The following theorem gives criterion for K-Riemnn integrbility. Theorem 8. Afunctionf :[, b] R is K-Riemnn integrble on [, b] if nd only if for every ε>0 there exists prtition π P[,b] K such tht U K f,π L K f,π <ε. Proof. Let ε>0nd choose prtition π P[,b] K tht stisfies the bove condition. Then, since we hve ftd K t U K f,π, 0 ftd K t nd L K f,π Since this inequlity holds for every ε>0, ftd K t U K f,π ftd K t U K f,π L K f,π <ε. ftd K t = ftd K t. Conversely, suppose tht f is K-Riemnn integrble. Given ny ε>0, there re prtitions π 1,π P[,b] K such tht U K f,π 1 < ftd K t + ε, L Kf,π > ftd K t ε.
Vol. 91 017 On K-Riemnn integrl nd the Hermite Hdmrd 433 Now, let π := π 1 π be the common refinement. Keeping in mind tht the K-Riemnn integrbility of f mens ftd Kt = ftd Kt we cn write U K f,π L K f,π U K f,π 1 L K f,π = U K f,π 1 ftd K t + ftd K t L K f,π < ε + ε = ε. Using the bove theorem we cn esily obtin the following Corollry 9. Afunctionf :[, b] R is K-Riemnn integrble on [, b] if nd only if for every sequence {π n } n N P[,b] K, π n =t n 0,tn 1,...,tn k n such tht mx t n j t n j 1 n 0, 1 j k n [ ] nd for ny choice s n j t n j 1,tn j of the prtition π n we hve ftd K t = lim K k n n j=1 f s n j t n j t n j 1. Proposition 10. Let K 1 K be subfields of R. If function f :[, b] R is K -Riemnn integrble then it is lso K 1 -Riemnn integrble, nd Proof. Let π n = such tht ftd K1 t = ftd K t. t n 0,tn 1,...,tn k n P K1 [,b], n N be n rbitrry sequence mx t n j t n j 1 n 0. 1 j k n By the K -Riemnn integrbility, for ny choice s n j prtition π n we hve ftd K t = lim k n n j=1 f s n j t n j Due to the rbitrriness of π n P K1 [,b] we infer tht ftd K1 t = lim k n n j=1 f s n j t n j [t n j 1,tn j ] K1 of the t n j 1. t n j 1.
434 Andrzej Olbryś AEM As n immedite consequence of the bove proposition we obtin the following. Corollry 11. If function f :[, b] R is Riemnn integrble in the usul sense, then for n rbitrry field K R f is K-Riemnn integrble, moreover, ftd K t = ftdt. Exmple 1. Let K 1 K, K 1 K be two subfields of R. Consider the following function f :[, b] R fx ={ 0, x [, b]k1 1, x [, b] K \ [, b] K1. It is esy to observe tht f is K 1 -Riemnn integrble, nd ftd K 1 t =0.On the other hnd for every prtition π P K [,b] \PK1 [,b] one cn check tht S K π, f =1, nd L K π, f =0. Therefore, 0= ftd K t ftd K t =1. Observe tht if we replce in the formul on f the set K 1 by the set D of didic numbers from the intervl [0, 1] i.e. D := {x [0, 1] x = k } n,k Z, n N, then we obtin n exmple of function which is non-k-riemnn integrble for ny subfield K R. 3. Properties of the K-Riemnn integrl We strt our investigtion with the following. Proposition 1. If f :[, b] R is function such tht f [,b]k is monotone then it is K-Riemnn integrble on [, b]. Proof. Assume tht f [,b]k is monotonic incresing, mening tht fx fy, for x y, x, y [, b] K. Fix n rbitrry sequence of prtitions π n = t n 0,tn 1,...,tn k n P[,b] K, n N,
Vol. 91 017 On K-Riemnn integrl nd the Hermite Hdmrd 435 where mx t n j t n j 1 n 0. 1 j k n Since f [,b]k is incresing, for ll j {1,...,k n } M j := sup ft =ft j, m j := inf ft =ft j 1. t [t j 1,t j] K t [t j 1,t j] K Hence, summing telescoping series, we get Uf,π n Lf,π n = k n j=1 M j m j t n j t n j 1 k n mx t n j t n j 1 [ft j ft j 1 ] 1 j k n j=1 = mx t n j t n j 1 [fb f]. 1 j k n It follows tht Uf,π n Lf,π n 0sn nd Corollry 9 implies tht f is K-Riemnn integrble. The proof for monotonic decresing function f is similr. In our next result we use well-known fct from mthemticl nlysis tht every uniformly continuous function on set A R n cn be uniquely extended onto cla to continuous function see for instnce [4] p. 06. Proposition 13. If f :[, b] R is uniformly rdilly K-continuous, then it is K-Riemnn integrble on ny subset [c, d] [, b]. Proof. Fix rbitrry c, d [, b], c < d. From Proposition 6 we infer tht f [c,d]k is uniformly continuous. Since cl[c, d] K =[c, d], there exists unique continuous function g cd :[c, d] R such tht g cd t =ft, t [c, d] K. On ccount of Corollry 11 f is K-Riemnn integrble, moreover, d c ftd K t = d c g cd tdt. In the sequel we will use the following well-known theorems see [8] p.147 ctully these theorems were proved for Jensen-convex functions, but the proof in our cse runs without ny essentil chnges. Theorem 14. Let I R be n open intervl, nd let f : I R be K-convex function. Then for rbitrry, b I, < b the function f [,b]k is uniformly continuous.
436 Andrzej Olbryś AEM Theorem 15. Let I R be n open intervl, nd let f : I R be K-convex function. Then for rbitrry, b I, < b there exists unique continuous function g b :[, b] R such tht g b x =fx, x [, b] K. The function g b stisfies the inequlity g b + b g bx+g b y, for every x, y [, b], in prticulr g b is convex function. Now, we clculte n integrl of K-liner function. Note tht such function cn be discontinuous t every point nd non-mesurble in the Lebesgue sense see [8], so the usul Riemnn integrl my not exist. Proposition 16. Let f : R R be K-liner function. Then it is K-Riemnn integrble on every intervl [, b], moreover, + b ftd K t = f b. Proof. Suppose tht f is K-liner function. On ccount of Proposition 13 ndtheorem14itisk-riemnn integrble on every intervl [, b]. Consider the following sequence of prtitions: π n := t n 0,tn 1,...,tn n, where, t n j := + j b, j =0, 1,...,n. n From Corollry 9 we obtin n ftd K t = lim f t n 1 n j b n j=1 = lim f nn +1 1 n + b b n n n = lim f + n +1 b b n n [ = lim f+ n +1 ] fb b n n [ ] b + b = f+f b =f b. Now, we record some bsic properties of K-Riemnn integrtion. We omit the proofs of these properties becuse they run in similr wy s for the usul Riemnn integrl.
Vol. 91 017 On K-Riemnn integrl nd the Hermite Hdmrd 437 Theorem 17. Let f,g be K-Riemnn integrble on [, b] nd let c, d R. Then i the function cf + dg is K-Riemnn integrble on [, b], moreover, [cft+dgt]d K t = c ftd K t + d gtd K t. ii If ft 0, t [, b] K, then ftd Kt 0, moreover,iff is rdilly K-continuous on [, b] nd ft > 0, t [, b] K then ftd K t>0. iii The bsolute vlue f is K-Riemnn integrble on [, b] nd ftd K t ft d K t. Theorem 18. Suppose tht f :[, b] R nd c [, b] K.Thenf is K-Riemnn integrble on [, b] if nd only if it is K-Riemnn integrble on [, c] nd [c, b]. Moreover, in tht cse, ftd K t = c ftd K t + c ftd K t. Proof. Assume tht f is K-Riemnn integrble on [, b]. Then, given ε>0 there is prtition π P[,b] K such tht U K f,π L K f,π <ε. Let π be the refinement of π obtined by dding c to the endpoints of π. Then π = π 1 π, where π 1 := π [, c] K, π := π [c, b] K. Obviously, π 1 P[,c] K nd π P[c,b] K, moreover, U K f,π =U K f,π 1 +U K f,π, Lf,π =L K f,π 1 +L K f,π. It follows tht U K f,π 1 L K f,π 1 =U K f,π L K f,π [U K f,π L K f,π ] U K f,π L K f,π <ε, which proves tht f is K-Riemnn integrble on [, c]. Exchnging π 1 nd π, we get the proof for [c, b]. Conversely, if f is K-Riemnn integrble on [, c] nd [c, b] then there re prtitions π 1 P[,c] K nd π P[c,b] K such tht U K f,π 1 L K f,π 1 < ε, U Kf,π L K f,π < ε.
438 Andrzej Olbryś AEM Let π := π 1 π. Then U K f,π L K f,π =U K f,π 1 L K f,π 1 +U K f,π L K f,π <ε, which proves tht f is K-Riemnn integrble on [, b]. Finlly, if f is K-Riemnn integrble, then with prtitions π, π 1,π bove, we hve Similrly, ftd K t U K f,π =U K f,π 1 +U K f,π <L K f,π 1 +L K f,π +ε < c ftd K t + c ftd K t + ε. ftd K t L K f,π =L K f,π 1 +L K f,π >U K f,π 1 +U K f,π ε > c Since ε>0 is rbitrry, we see tht ftd K t = ftd K t + c c ftd K t + c ftd K t ε. ftd K t. s Remrk 19. Observe tht for K-liner function f : R R, where K R nd point c = α +1 αb, b, where α 0, 1 we hve c ftd K t ftd K t ftd K t = 1 fα b αf b b. c Therefore, it cn hppen tht for some α 0, 1 \ K the bove expression is different from zero. 4. Connections between the rdil K-derivtive nd the K-Riemnn integrl In 006 Z. Boros nd Zs. Páles in [1] introduced nd exmined the notion of rdil K-derivtive of mp t point in the given direction. Definition 0. A mp f : I R I stnds for n open intervl is sid to hve rdil K-derivtive t point x I in the direction u R provided tht there exists finite limit
Vol. 91 017 On K-Riemnn integrl nd the Hermite Hdmrd 439 D K fx, u := lim K + r 0 fx + ru fx. r We will sy tht f is rdilly K-differentible t point x whenever D K fx, u does exist for every u R. A function f : I R is termed rdilly K- differentible if f is rdilly K-differentible t every point x R. It is known tht ech K-convex function f : I R is rdilly K-differentible. In prticulr, such is every K-liner function : R R with D K x, u =u, x,u R. On the other hnd, if function f : I R is differentible in the usul sense t point x I then it is rdilly K-differentible t x with D K fx, u =f xu, for u R. We hve the following reltionship between the rdil K-derivtive nd the K-Riemnn integrl. Theorem 1. Suppose tht f :[, b] R is K-Riemnn integrble on ech subset of the form [, x] K, for ny x, b]. Let us define the function F : [, b] R by the formul F x := x ftd K t. Then, if f is rdilly K-continuous t point x, b] then F is rdilly K-differentible t x in the direction x, moreover, D K F x, x =fxx. Proof. Fix x [, b] ndε>0 rbitrrily. For α K +, since x [, x + αx ] K, on ccount of Theorem 18 nd condition iii from Theorem 17 we obtin α fxx x+αx F x+αx F x = = 1 α 1 x+αx α x x+αx = 1 α x Let α K + be so smll tht ft fx < ftd K t x ftd Kt x+αx x ftd K t 1 α ft fxd K t 1 α 1 x+αx α x fxd K t fxd K t x+αx x ft fx d K t. ε x, for t [x, x + αx ] K.
440 Andrzej Olbryś AEM Then, 1 α x+αx x 1 α x+αx x ft fx d K t ε x d Kt = 1 ε αx α x = ε. Now, we re in position to prove the following chrcteriztion of K- convex functions. Theorem. Let I R be n intervl, nd let f : I R be K-convex function. Then, for every, x I, we hve fx =f+ 1 x D K ft, x d K t. x Proof. Tke rbitrry, x I, x, sy < x. Since f is K-convex function, on ccount of Theorem 15 there exists uniformly continuous, convex function g :[, x] R such tht ft =gt, t [, x] K. Therefore, for t [, x] K we get ft + αx ft D K ft, x = lim K + α 0 α gt + αx gt =x lim =x g K + α 0 αx +t. It follows from the bove formul nd from the fundmentl theorem of clculus for the usul Riemnn integrl tht 1 x x D K ft, x d K t = g x +tdt = gx g =fx f, which ws to be proved. 5. Hermite Hdmrd inequlities There re mny inequlities vlid for convex functions. Probbly two of the most well-known ones re the Hermite Hdmrd [3, 5 8, 11, 16] inequlities. + b f 1 ftdt f+fb, < b. b They ply n importnt role in convex nlysis. In the literture one cn find their vrious generliztions nd pplictions. For more informtion on this type of inequlities see the book [3] nd the references therein. We just note
Vol. 91 017 On K-Riemnn integrl nd the Hermite Hdmrd 441 here tht first Hermite [7] published these inequlities with some importnt pplictions nd then, 10 yers lter, Hdmrd [5] rediscovered their left-hnd side. It turns out tht ech of the two sides of in fct chrcterizes convex functions. More precisely, if I is n intervl nd f : I R continuous function whose restriction to every compct subintervl [, b] verifies the lefthnd side then f is convex. The sme works when the left-hnd side is replced by the right-hnd side. More generl results re given by Rdo [13]. Now, we re in position to prove our min result. The following theorem estblishes the Hermite Hdmrd inequlities for K-convex functions. Theorem 3. Let I R be nonempty open intervl nd let f : I R be K-convex function. Then for rbitrry, b I, < b the inequlities + b f 1 ftd K t f+fb, < b, 3 b hold. Proof. Let, b I, < b be rbitrrily fixed. It follows from Theorem 15 tht there exists unique continuous nd convex function g b :[, b] R such tht g b x =fx, x [, b] K. Since g b is convex, it stisfies the clssicl Hermite Hdmrd inequlities, nmely + b + b f = g b 1 g b tdt g b+g b b b However, on ccount of Corollry 11 which finishes the proof. g b tdt = ftd K t, = f+fb. Since in the proof of the bove theorem we used the clssicl Hermite Hdmrd inequlities, we cn not sy tht it is more generl result. Therefore, now we give nother proof without using these inequlities. Proof. To prove the right-hnd side of 3 observe tht fb f fx f+ x, x [, b] K, b so, integrting the bove inequlity over [, b] nd dividing by b we get 1 b fxd K x f+fb.
44 Andrzej Olbryś AEM To obtin the left-hnd side of we use the following esy-to-prove expression ftd K t =b 1 0 fs +1 sbd K s. Using the bove formul nd the Jensen-convexity of f we get 1 b b ftd Kt = 1 +b b ftd K t + b +b ftd K t = 1 1 0 [ f +b tb + f ] +b+tb d K t f +b. It turns out, tht s with convex functions, in the clss of uniformly rdilly K-continuous functions ech of the inequlities 3 is equivlent to K-convexity. Nmely, the following theorem holds true. Theorem 4. If function f : I R is uniformly rdilly K-continuous nd, for ll elements <bof I, stisfies either the inequlity + b f 1 ftd K t, b or then it is K-convex. 1 b ftd K t f+fb, Proof. Suppose tht f stisfies the first inequlity for the second inequlity the proof runs in similr wy. It is enough to prove tht for every, b I, < b unique extension g b of f [,b]k onto [, b] to continuous function is convex. To see it, fix rbitrrily c, d [, b],c < d. There exist sequences {c n } n N nd {d n } n N such tht c n,d n [, b] K,c n <d n,n N, nd lim n c n = c, lim d n = d. n Since c n,d n [, b] K, the extension g cnd n onto [c n,d n ] to continuous function by virtue of uniqueness stisfies the condition g b t =g cnd n t =ft, t [c n,d n ] K, n N. By the ssumption for ll n N cn + d n cn + d n 1 dn 1 dn g b = f ftd K t = g b tdt. d n c n c n d n c n c n Tking limits s n gives c + d g b 1 d g b tdt. d c c
Vol. 91 017 On K-Riemnn integrl nd the Hermite Hdmrd 443 We hve shown tht continuous function g b stisfies the left-hnd side of the clssicl Hermite Hdmrd inequlities, so s we know, it is convex. Due to the rbitrriness of, b I we infer tht f is K-convex, which finishes the proof. Open Access. This rticle is distributed under the terms of the Cretive Commons Attribution 4.0 Interntionl License http://cretivecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthors nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde. References [1] Boros, Z.: Zs. Páles, Q-subdifferentil of Jensen-convex functions. J. Mth. Anl. Appl. 31, 99 113 006 [] Dróczy, Z., Páles, Z.: Convexity with given infinite weight sequences. Stochstic 11, 5 1 1987 [3] Drgomir, S.S., Perce, C.E.M.: Selected Topics on Hermite Hdmrd Inequlities nd Applictions. Victori University, RGMIA Monogrphs 00 [4] Giguint, M., Modic, G.: Mthemticl Anlysis. Liner nd Metric Structures nd Continuity. Springer, New York 007 [5] Hdmrd, J.: Étude sur les properiétés entiéres et en prticulier dúne fonction considerée pr Riemnn. J. Mth. Pures Appl. 58, 171 15 1893 [6] Hrdy, G., Littlewood, J.E., Poly, G.: Inequlities. Cmbridge University Press, Cmbridge 1951 [7] Hermite, C.: Sur deux limites dúne intégrle définie. Mthesis 3, 8 1883 [8] Kuczm, M.: An introduction to the theory of functionl equtions nd inequlities. Birkhäuser, Bsel 009 [9] Kuhn, N.: A note on t-convex functions. Gen. Inequl. 4, 69 76 1984 [10] Kurtz, D.S., Kurzweil, J., Swrtz, C.: Theories of Integrtion: The Integrls of Riemnn, Lebesgue, Henstock Kurzweil, nd McShne, Volume 9 of Series in Rel Anlysis. World Scientific 004 [11] Niculescu, C.P., Persson, L.E.: Convex Functions nd their Applictions, A Contemporry Approch CMS Books in Mthemtics, vol. 3. Springer, New York 006 [1] Páles, Zs: Problem, Report of Meeting, The Thirteenth Ktowice-Debrecen Winter Seminr. Ann. Mth. Sil. 7, 107 15 013 [13] Rdo, T.: On convex functions. Trns. Am. Mth. Soc. 37, 66 85 1935 [14] Riemnn, G.F.B.: In: H. Weber ed. Gesmmelte Mthemtische Werke. Dover Publictions, New York 1953 [15] Riemnn, G.F.B.: In: K. Httendorff ed. Prtielle Differentilgleichungen und deren Anwendung uf physiklische Frgen. Vieweg, Brnschweig 1869 [16] Roberts, A.W., Vrberg, D.E.: Convex Functions. Acdemic Press, New York 1973
444 Andrzej Olbryś AEM Andrzej Olbryś Institute of Mthemtics University of Silesi Bnkow 14 40-007 Ktowice Polnd e-mil: ndrzej.olbrys@us.edu.pl Received: September 7, 016 Revised: Jnury 5, 017