New Hermite-Hadamard and Jensen Type Inequalities for h Convex Functions on Fractal Sets

Transcription

1 Revist Colomin de Mtemátics Volumen 56, págins New Hermite-Hdmrd nd Jensen Type Inequlities for h Convex Functions on Frctl Sets Nuevs desigulddes del tipo Hermite-Hdmrd y Jensen pr funciones h-convexs sore conjuntos frctles Miguel Vivs,,, Jorge Hernández, Nelson Merentes 3 Escuel Superior politécnic del Litorl ESPOL, Guyquil, Ecudor Universidd Centroccidentl Lisndro Alvrdo, Brquisimeto, Venezuel 3 Universidd Centrl de Venezuel, Crcs, Venezuel Astrct. In this pper, some new Jensen nd Hermite-Hdmrd inequlities for h-convex functions on frctl sets re otined. Results proved in this pper my stimulte further reserch in this re. Key words nd phrses. generlized convexity, h-convex functions, Frctl sets, Hermite-Hdmrd type inequlity, Jensen inequlity. Mthemtics Suject Clssifiction. 53C, 53C4. Resumen. En este rtículo, se otienen lguns nuevs desigulddes del tipo Jensen y Hermite - Hdmrd pr funciones h-convexs sore conjuntos frctles. Los resultdos prodos en este rtículo pueden estimulr futurs investigciones en est áre. Plrs y frses clve. convexidd generlizd, funciones h-convexs, conjuntos frctles, desiguldd del tipo Hermite Hdmrd, Desiguldd del tipo Jensen. 45

2 46 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES. Introduction Frctls hve een known for out more thn century nd hve een oserved in different rnches of science. But it is only recently pproximtely in the lst forthy yers tht they hve ecome suject of mthemticl study. The pioneer of the theory of frctls ws Benoit Mndelrot. His ook Frctls: Form, Chnce nd Dimension first ppered in 977, nd second, enlrged, edition ws pulished in 98. Since tht time, serious rticles, surveys, populr ppers, nd ooks out frctls hve ppered y the dozen. Mndelrot in [7] defined frctl set is one whose Husdorff dimension exceeds strictly its topologicl dimension. Also, Yng in [3] estlished the numericl α sets, where α is the dimension of the considered frctl. For more detils out frctl sets see for instnce [6, 7, 8, 3] nd references therein. It is well known tht modern nlysis directly or indirectly involves the pplictions of convexity. Due to its pplictions nd significnt importnce, the concept of convexity hs een extended nd generlized in severl directions. The concept of convexity nd its vrint forms hve plyed fundmentl role in the development of vrious fields. Convex functions re powerful tools for proving lrge clss of inequlities. They provide n elegnt nd unified tretment of the most importnt clssicl inequlities. A significnt generliztion of convex functions is tht of h-convex functions introduced y Snj Vrošnec in [8]. There re mny results ssocited with convex functions in the re of inequlities, two of those re: the Jensen inequlity nd the Hermite-Hdmrd inequlity, which occur widely in the mthemticl literture. In this pper, we will estlish some new integrl inequlities of Hermite-Hdmrd type for h-convex functions. The following definition is well known in the literture s convex function: function f : I R R is sid to e convex if ftx + ty tfx + tfy holds for ll x, y I nd t [, ]. The convexity of functions nd their generlized forms ply n importnt role in mny fields such s Economic Science, Biology, Optimiztion. In recent yers, severl extensions, refinements, nd generliztions hve een considered for clssicl convexity [, 5, 4, 6, 8, 9,, 6, 8]. The clssicl Jensen inequlity is contined in the following theorem. Theorem. See []. Let f : I R R e convex function over I. Then for every x i I, t i [, ], i =,,..., n, nd n t i =, we hve n f t i x i t i fx i. Volumen 5, Número, Año 6

3 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS47 Jensen s inequlity is sometimes clled the king of inequlities since it implies the whole series of other clssicl inequlities e.g. those y Hölder, Minkowski, Beckench-Dresher nd Young, the rithmetic-geometric men inequlity etc.. Jensen s inequlity for convex functions is proly one of the most importnt inequlities which is extensively used in lmost ll res of mthemtics, especilly in mthemticl nlysis nd sttistics. For comprehensive inspection of the clssicl nd recent results relted to the inequlity the reder is referred to [, 5, 7, 9]. One of the gols of this rticle is to estlish Jensen-type inequlity for generlized h-convex functions. It is well-known tht one of the most fundmentl nd interesting inequlities for clssicl convex functions is tht ssocited with the nme of Hermite- Hdmrd inequlity which provides lower nd n upper estimtions for the integrl verge of ny convex functions defined on compct intervl, involving the midpoint nd the endpoints of the domin. More precisely: Theorem. See []. Let f e convex function over [, ], <. If f is integrle over [, ], then + f + f f f x dx. The ove inequlity ws firstly discovered y Hermite in 88 in the journl Mthesis see Mitrinović nd L cković [9]. But, this eutiful result ws nowhere mentioned in the mthemticl literture nd ws not widely known s Hermite s result see Klričić et l. []. For more recent results which generlize, improve, nd extend the clssicl Hermite-Hdmrd inequlity, see for instnce [5, 4, 5], nd references therein. The Hermite-Hdmrd inequlity hs severl pplictions in nonliner nlysis nd the geometry of Bnch spces, see [, 3]. In the present pper,we re concerned with n nlogous of Theorem for h-convex functions on frctl set. Let us recll two importnt definitions of generlized convex functions. Definition.3 See [9]. We shll sy tht function f : I R R is Godunov-Levin function or f QI if f is non negtive nd for ech x, y I nd t, we hve ftx + ty fx t + fy t. Definition.4 See [3]. Let s, ]. A function f :, ], ] is clled s convex function in the second sense, or f K s if for ech x, y, ] nd t [, ]. ftx + ty t s fx + t s fy Revist Colomin de Mtemátics

4 48 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES It is cler tht,for s =, s convexity reduces to ordinry convexity of functions defined on, ]. In the yer 999, Drgomir [5] proved vrint of the Hermite-Hdmrd inequlity, for s-convex functions in the second sense. Theorem.5 See [5]. Let f :, ], ] s convex function in the second sense,with s, ], nd,, ], <. If f L [, ], then we hve + s f fxdx f + f. s + In the yer 7, Vro snec, [8], defined the following so-clled h-convex function: Definition.6. Let h : J R e non-negtive,non-identiclly zero function, defined on n intervl J R, with, J. We shll sy tht function f : I R, defined on n intervl I R, is h-convex if f is non negtive nd this inequlity holds for ll t, nd x, y I. f tx + t y h t f x + h t f y When ht = t, this definition coincides with the ordinry convex function. If ht = t s with < s, the coincidence is with the s convex functions, nd if ht = /t this coincides with the Godunov-Levin type of generlized convex function. For other recent results nd properties of the clss of h-convex functions see [, 6, 3, 4, ]. In this rticle motivted nd inspired y the ongoing reserch in the field [3, 4,, ],we estlish new Hermite-Hdmrd nd Jensen type inequlities for h-convex functions on frctl sets. The rticle is orgnized s follows: In section we stte the opertions with rel line numers on frctl sets nd we recll some definitions nd preliminry fcts of frctionl clculus theory which will e used in this pper, lso we introduce the definition of h-convexity on frctl sets. In secction 3,we estlish the min results of the rticle: the generlized Jense s inequlity nd generlized Hermite-Hdmrd s inequlty for generlized h-convex functions. In section 4 we give some pplictions/exmples to illustrte.. Preliminries nd Bsic Results Recently, the theory of Yng s frctionl set of elements sets ws introduced s follows: For < α we hve the following α type sets. Volumen 5, Número, Año 6

5 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS49 Z α = { α, ± α, ± α,..., ±n α,...} Q α = {/ α : α Z α, α Z α, α α } I α = {m α / α : α Z α, α Z α, α α } R α = Q α I α For α, α, c α R α the following properties hold:. α + α R α y α α R α. α + α = α + α = + α = + α c. α + α + c α = α + α + c α d. α α = α α = α = α e. α α c α = α α c α f. α + α = α + α = α y α α = α α = α If α α is non negtive we sy α is greter thn or equl to α, or α is less thn or equl to α, nd we write α α or α α, respectively. If there is not possiility tht α = α then we write α > α o α < α. Next we recll some definitions nd some fcts of frctionl clculus theory on R α which will e used in this pper. Definition.. Let f : R R α e mpping. We sy tht f is locl frctionl continuous t x R, if for ll ɛ > exists δ > such tht x x < δ = f x f x < ɛ α If f is locl frctionl continuous in ech point of n intervl,, we sy tht f is locl frctionl continuous in, nd we write f C α,. Definition.. The locl frctionl derivtive of f of order α t x = x is defined y f α x = dα fx α fx fx dx α = lim x x x x α x=x where fx fx = Γ α + fx fx. Definition.3. Let f C α [, ]. Then the locl frctionl integrl of order α of f is defined y f = Γ + α I α = Γ + α lim f x dx α t N f t i t i α Revist Colomin de Mtemátics

6 5 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES where t i = t i+ t i, t = mx{ t,..., t N }, nd [t i, t i+ ], i =,,..., N, with = t < t < < t N =, is prtition of [, ]. nd If for ech x [, ] there exists I α Here, it follows I α f = if = I α f = I α f if <. Also we hve the property of chnge of vriles. f, then we write f I x α [, ]. Lemm.4. If g C α [, ] nd f C α [g, g] then gi α g f = Γ + α = Γ + α g g = I α f g g. f x dx α f gt g t dt α In [], Mo nd Sui considered the following denfition of generlized convexity on frctl set. Definition.5. Let f : I R α, with < α. For ny x x in I nd t [, ], we sy tht f is generlized convex function on I if holds. ftx + tx t α fx + t α fx In [], the definition of s convex functions on frctl sets ws estlished s follows: Definition.6. A function f : R + R α is sid to e generlized s convex < s < in the second sense, if ft x + t x t sα fx + t sα fx for ll x, x R + nd ll t, t > with t + t =. With this, they otin the following results. Theorem.7. Let f : I R α e generlized convex function. Then for ech x i [, ] nd t i [, ] with i =,,..., n we hve n f t i x i t α i fx i. Volumen 5, Número, Año 6

7 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS5 Theorem.8. Let f I x α [, ] e generlized convex function on [, ] with <. Then + f Γ + α α fxdx α f + f α. Next we give our definition of generlized h -convex functions on frctl set. Definition.9. Let h : J R α e non-negtive function nd h, defined over n intervl J R nd such tht, J. We sy tht f : I R α, defined over n intervl I R, is h-convex if f is non negtive nd we hve for ll t, nd x, y I. f tx + t y h t f x + h t f y 3 We cn see tht if ht t α, like ht = t kα, where < k then ny non-negtive nd convex function f : I R α is h-convex on R α. In [] we cn find nother exmple of such functions. Exmple.. Let < s <, h :, R α defined y ht = t sα, t, nd α, α, c α R α. For x R +, define fx = { α, si x = α x sα + c α, si x > 3. Min Results In this section, we estlish our min results. Theorem 3.. Let t,..., t n e positive rel numers. If h : J R α is nonnegtive function, h, supermultiplictive defined over n intervl J R nd such tht, J, nd let f : I R α e function defined over n intervl I R, h convex, nd x,..., x n I, then where = n t i. f t i x i ti h f x i 4 Proof. The proof is y induction. If n =, the desired inequlity is otined from the definition of h-convex function 3 with t = t T, t = t T, x = x nd y = x. Assume tht for n, where n is ny positive integer, the inequlity 4 is lso true. Revist Colomin de Mtemátics

8 5 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES Then, we see tht f t i x i = f = f t n n t i x i x n + t n x n + n t i x i Using the definition.9 in the right-hnd side of the previous inequlity, we hve n tn Tn t i f t i x i h f x n + h f x i. Now, s we hve ssumed tht 4 holds for n we otin n tn Tn ti f t i x i h f x n + h h tn = h n f x n + Tn h h. ti Further, since h is supermultiplictive function, we cn see Tn ti Tn t i h h h = h using this fct we otin f n tn t i x i h f x n + ti h f x i = ti, f x i f x i. ti h f x i. The ove inequlity holds y the result for n= nd the induction hypothesis. Remrk 3.. If ht = t α we hve f t i x i ti α f x i nd if we put λ i = t i /, i =,..n then n f λ i x i λ α i f x i Volumen 5, Número, Año 6

9 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS53 nd this coincides with the result demonstrted y Mo nd Sui in [] out generlized convex function over frctl set. In the sme wy if ht = t sα, < s <, we hve sα ti f t i x i f x i nd if λ i = t i /, i =,..n then n f λ i x i λ sα i f x i corresponding to generlized s convex functions over frctl sets. The next result involves n integrl inequlity of Hermite-Hdmrd type. Theorem 3.3. Let h : J R α e non-negtive integrle function, h, defined over n intervl J R nd such tht, J nd f : I R α e n h convex, non-negtive nd integrle function,, I with <. Then α h/γ + α f + α I α f 5 Proof. Note tht f α f I α h. t + t + t + t = t + t + t + t = + for ll t [, ]. And s f is n h-convex function, we hve + f h / f t + t + h / f t + t Thus, integrting oth sides, we get + f dt α h / Now, we note tht = h / f t + t + f t + t. f t + t dt α +h / f t + t dt α = α α f t + t dt α. fx dx α Revist Colomin de Mtemátics

10 54 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES nd f t + t dt α = α nd with this we hve + f dt α h / α α from which it follows tht α α h / Γ + α f which corresponds to the left inequlity in 3.3. fx dx α fx dx α + α I α f We know tht for ny x [, ] there exists t [, ] such tht x = t + t. With this fct nd the h-convexity of f, we cn write f x dx α = α f t + t dt α n so we otin α h t f + h t f dt α = α f h t dt α + f = α f α h t dt α + f = α α f + f h t dt α α I α f α f + f I α h h t dt α which corresponds to the right-hnd side of 3.3, nd we cn conclude α h/γ + α f + α I α f h t dt α This complete the proof. f α f I α h. Remrk 3.4. Oserve tht if ht = t nd α = then, Γ + α = Γ =, h/ = /, I α h = h t dt α = Γ + α Volumen 5, Número, Año 6

11 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS55 nd I α f = consequently we get from 3.3. fxdx Remrk 3.5. If ht = t s with s, ] y α =, then Γ + α = Γ = y h/ = / s we get nd therefore I α h = Γ + α I α f = s f + h t dt α = s + fxdx fxdx f + f s + nd it corresponds to the result otined in [5] for s convex functions in the second sense. Theorem 3.6. Let h, h : J R α e two non-negtive functions nd h, h, defined over n intervl J R nd such tht, J, moreover h I x α [, ], h I x α [, ] nd h h I x α [, ]. Let f e n h convex function, nd g n h convex function, oth non-negtive on R α,, I, < nd such tht fg I x α [, ]. Then α α I α fg M, I α h h + N, I α h h t 6 where M, = fg + α fg nd N, = fg + α fg. Proof. Since f is h convexfunction, nd g is h convexfunction, nd for ech x [, ] exists t [, ] such tht x = t + t, we hve nd ft + t h tf + h tf gt + t h tg + h tg. Further, since f nd g re non-negtive, then ft + tgt + t h tf + h tfh tg + h tg Revist Colomin de Mtemátics

12 56 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES = h th tfg + h th tfg +h th tfg + h th tfg integrting over [, ] oth sides of the inequlity, we otin ft + tgt + tdt α fg + fg + fg + fg h th tdt α h th tdt α h th tdt α h th tdt α = fg α fg + fg α fg From the proof of the previous Theorem we hve Then α α where nd f t + t dt α = α α h th tdt α h th tdt α. fx dx α. fxgx dx α M, I α h h + N, I α h h M, = fg + α fg N, = fg + α fg. Theorem 3.7. Let h, h : J R α e two non negtive functions nd h, h, defined over n intervl J R nd such tht, J, nd h h I x α [, ] Let f n h convex function, nd g n h convex function, oth non-negtive over R α,, I, < such tht fg I x α [, ]. Then α α h /h / f Γ + α g α I α fg Volumen 5, Número, Año 6

13 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS57 M, Γ + α I α h h + N, Γ + α I α h h where M, = fg + fg nd N, = fg + fg. Proof. Let, I with <. Then we cn write + = t + t + t + t for ll t [, ]. In consequence, + + t + t t + t f g = f + t + t t + t g +. Since f is h convex nd g is h convex, we hve + + f g h /[f t + t + f t + t] h /[g t + t + g t + t]. Using distriutive property we get + + f g h /h /[f t + t g t + t + f t + t g t + t + f t + t g t + t + f t + t g t + t]. Forming groups with the terms we hve + + f g h /h /[f t + t g t + t 7 +f t + t g t + t]+ h /h /[f t + t g t + t +f t + t g t + t]. Agin, using the h convexity nd h convexity of f nd g respectively, nd distriuting the products in the second term of the sum in the previous inequlity, we cn oserve f t + tg t + t + f t + t g t + t h tfh tg + h tfh tg + h tfh tg + h tfh tg + h tfh tg + h tfh tg + h tfh tg + h tfh tg. Revist Colomin de Mtemátics

14 58 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES Now, we grouped the terms conveniently f t + tg t + t + f t + t g t + t = {h th t + h th t}m, + {h th t + h th t}n, where nd M, = fg + fg N, = fg + fg. In consequence, the inequlity 7 tkes the form + + f g h /h /[f t + t g t + t 8 Oserve the following integrls + f g +f t + t g t + t] +h /h / [h th t + h th t] M, + + [h th t + h th t] N,. dt α = α f + g + h th tdt α = α h th tdt α, h th tdt α = α h th tdt α nd mking the sustitution x = t + t we get f t + t g t + tdt α = α α nd with the sustitution x = t + t we hve f t + t g t + tdt α = α, fxgxdx α fxgxdx α. Volumen 5, Número, Año 6

15 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS59 So, with this chnges nd integrting oth sides of the inequlity 8 over [, ] we otin + + α f g nd follows h /h / α α + h /h /{M, α + N, α α + α h /h / f M, + g h th tdt α } + α h th tdt α + N, fxgxdx α h th tdt α fxgxdx α h th tdt α. Using the frctionl integrl definition, this inequlity cn e written s α α h /h / f + g + Γ + α α I α fg M, Γ + α I α h h + N, Γ + α I α h h nd this is the desired result. Remrk 3.8. Clerly, if h t = h t = t nd α = we otin + f g + which is the Theorem given y Pchptte in [4]. fxgxdx 6 M, + N, 3 Remrk 3.9. For s convex functions in second sense lso we get result showed y Kircmci et l. in [5]. Mking h t = t,h t = t s with α = nd s, ] we otin + s f g + fxgxdx M, N, + s + s + s +. Revist Colomin de Mtemátics

16 6 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES 4. Exmples Exmple 4.. Let >, >, x. nd 3α + 3α α. Then +. Proof. Let fx = x 3α for x,. It is esy to see tht f is n h-convex function for hλ = λ α, for ny [, ],. Indeed t + t 3α = t 3α 3α + t 3α 3α t α 3α + t α 3α. Then in consequence it follows tht + f h/f + h/f + 3α = 3α + 3α 3α 3α + 3α α = h/ 3α + h/ 3α = h/ 3α + 3α α α = α 3 + hence therefore, +. + Exmple 4.. Let < α, <, <,with <, t [, ] nd fx = Lnx + nd ht = t + α then f is h-convex. Indeed, it follows tht nd therefore Ln + α Lnt + t + Ln + α Lnt + t + Ln + α + Ln + α Lnt + t + t + α Ln + α + t α Ln + α since for t, we hve t + y t. Volumen 5, Número, Año 6

17 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS6 Now, note tht in the Hermite Hdmrd inequlity 3.3 α h/γ + α f + α I α f with α = we get Ln f α f I α h therefore we otin the estimtes Ln Lnx + dx Ln + + Ln + t + dt Lnx + dx 7 Ln [ + + ]. 3 References [] M. Klričić Bkul nd J. Pečrić, Note of some hdmrd type inequlity, Journl of Inequlities in Pure nd Applied Mthemtics 5 4, 9 4. [] M. Bomrdelli nd S. Vro snec, Properties of h-convex functions relted to the hermite-hdmrd-fejér inequlities, Computers nd Mthemtics with Applictions 58 9, [3] W. W. Breckner, Stetigkeitsussgen für eineklsse verllgemeinerter konvexer funktionen in topologischen lineren räumen, Pu. Inst. Mth , 3. [4] S. S. Drgomir, Inequlities of hermite-hdmrd type for h-convex functions on liner spces, Proyecciones Journl of Mthemtics 3 5, [5] S. S. Drgomir nd S. Fitzptrick, The hdmrd.s inequlity for s-convex functions in the second sense, Demonstrtio Mth , [6] G. A. Edgr, Mesure, topology, nd frctl geometry, Springer-Verlg,New York, 99. [7] K. Flconer, The geometry of frctl sets, CmridgeUniversity Press,Cmridge, 985. Revist Colomin de Mtemátics

18 6 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES [8], Frctl geometry, John Wiley nd Sons, Chichester, 99. [9] E. Gounov nd V. Levin, Inequlities for functions of rod clss tht contins convex, monotone nd some other forms of functions. russin numericl mthemtics nd mthemticl physics russin, Numericl mthemtics nd mthemticl physics Russin. Moskov. Gos. Ped. Inst., Moscow , [], Nervenstv dlj funkcii sirokogo klss, soder z s cego vypuklye, monotonnye,i nekotorye drugie vidy funkcii, in : Vy cislitel, Mt. i. Mt. Fiz. Me zvuzov. S. Nu c. Trudov. MGPI. Moskv 3 985, [] J. L. W Jensen, Sur les fonctions convexes et le inequlitiés entre les vleurs moyennes, Act Mth. 3 96, [] Eder Kikinty, Hermite-hdmrd inequlity in the geometry of nch spces, Tesis doctorl, School of Engineering nd Science Fculty of Helth, Engineering nd Science Victori University,, The purpose of this thesis is to employ the Hermite-Hdmrd inequlity in studying the geometry of Bnch spces. [3] A. Kiliçmn nd W. Sleh, Notions of generlized s-convex functions on frctl sets, Journl of Inequlities nd Applictions. Spingeropen Journl 3 5. [4], Some generlized hermite-hdmrd type integrl inequlities for generlized s-convex functions on frctl sets, Advnces in Differences Equtions. Spingeropen Journl 3 5. [5] U. S. Kirmci, M. K. Bkul, M. F. Özdemir, nd J. Pečrić, Hdmrd type inequlities for s-convex functions, Appl. Mth. nd Comp 93 7, [6] M. A Ltif, On some inequlities for h-convex functions, Int. Journl of Mth. Anlysis 4, no. 3, [7] B. B. Mndelrot, The frctl geometry of nture, Mcmilln, New York, Ny, USA, 983. [8] N. Merentes nd S. Rivs, El desrrollo del concepto de función convex, XXVI Escuel venezoln de Mtemátics. Emlc - Venezuel, 3. [9] D. S. Mitrinović nd I. B. L cković, Hermite nd convexity, Aequtiones Mth , 5 3. [] D. S. Mitrinović nd J. Pečrić, Clssicl nd new inequlities in nlysis, Kluwer Acdemic Pulishers, Dordrecht /Boston /London, 993. Volumen 5, Número, Año 6

19 HERMITE - HADAMARD AND JENSEN INEQUALITIES FOR H-CONVEX FUNCTIONS63 [] H. Mo nd X. Sui, Generlized s-convex functions on frctl sets, ArXiv: v 8 4, 5 3. [] H. Mo, X. Sui, nd D. Yu, Generlized convex functions on frctl sets nd two relted inequlities, Astrct nd Applied Anlisys 8 4, 5 3. [3] M. Noor, Hermite-hdmrd integrl inequlities fot log ϕ convex functions, Nonliner Anisys Forum 3 8, no., 9 4. [4] B. G. Pchptte, On some inequlities for convex functions, RGMIA. Res.Rep.Coll.6 Coll. 6 3, 9 4. [5] J. E. Pečrić, F. Proschn, nd Y. L. Tong, Convex functions, prtil orderings, nd sttisticl pplictions, Acdemic Press, Inc., 99. [6] A. W. Roerts nd D. Vlerg, Convex functions, Pure nd Applied Mthemtics. Acdemic Press, 973. [7] S. Simi`c, On new converse of jensen s inequlity, Pulictions de L Intitut Mthémtique. Nouvelle serie 85 9, no. 99, 7. [8] S. Vro snec, On h convexity, J. Mth. Anl. Appl. 36 7, [9] L. Wng, X. M, nd L. Liu, A note on some new refinements of jensen s inequlity for convex functions, Journl of Inequlities in Pure nd Applied Mthemtics 9, no., 6 pp. [3] X. J. Yng, Advnced locl frctionl clculus n plictions, World Science, NY, USA,. Reciido en mrzo de 6. Aceptdo en septiemre de 6 Deprtmento de Mtemátics Fcultd de Ciencis Nturles y Mtemátics Escuel Superior politécnic del Litorl ESPOL Cmpus, Gustvo Glindo Km 3.5 Ví perimetrl, Guyquil, Ecudor e-mil: mjvivs@espol.edu.ec Deprtmento de Mtemátics Decnto de Ciencis y Tecnologí Universidd Centroccidentl Lisndro Alvrdo Av., Av.Morn Brquisimeto, Venezuel e-mil: mvivs@ucl.edu.ve Revist Colomin de Mtemátics

20 64 MIGUEL VIVAS, JORGE HERNÁNDEZ & NELSON MERENTES Deprtmento de Técnics Cuntittivs Universidd Centroccidentl Lisndro Alvrdo Decnto de Ciencis Económics y Empresriles Av., Av.Morn Brquisimeto, Venezuel e-mil: jorgehernndez@ucl.edu.ve Escuel de Mtemátics Universidd Centrl de Venezuel Fcultd de Ciencis Av. Los Chgurmos Crcs, Venezuel e-mil: nmerucv@gmil.com Volumen 5, Número, Año 6