A sandwich theorem and stability result of Hyers-Ulam type for harmonically convex functions

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1 Lecturas Matemáticas Volumen , páginas 5-8 ISSN A sandwich theorem and stability result o Hyers-Ulam type or harmonically conve unctions Un teorema del sándwich y un resultado de estabilidad de tipo Hyers-Ulam para unciones armónicamente conveas Mireya Bracamonte,3, José Giménez 2, Jesús Medina and Miguel Vivas-Cortez 3, Universidad Centroccidental Lisandro Alvarado,Venezuela 2 Universidad de los Andes, Venezuela 3 Escuela Superior Politécnica del Litoral ESPOL, Ecuador ABSRAC. We prove that the real unctions and g, deined on a real interval [a, b, satisy tg y + tg, t + ty or all, y [a, b and t [0, i there eists a harmonically conve unction h : [a, b R such that h g or all [a, b. We also obtain an approimate conveity result, namely we prove a stability result o Hyers-Ulam type or harmonically conve unctions. Key words: Harmonically conve unctions, Sandwich theorem, Hyers-Ulam. RESUMEN. Demostramos que las unciones reales y g, deinidas en un intervalo [a, b, satisacen tg y + tg t + ty para todo, y [a, b y t [0, si y sólo si eiste una unción armónicamente convea h : [a, b R tal que h g para cada [a, b. ambién obtenemos un resultado de aproimación convea, es decir, se demuestra un resultado de estabilidad del tipo de Hyers-Ulam para unciones armónicamente conveas.

2 6 Mireya Bracamonte et al. A sandwich theorem and stability result o Hyers-Ulam... Palabras clave: Funciones armónicamente conveas, teorema del sándwich, Hyers- Ulam. 200 AMS Mathematics Subject Classiication. 26A99,52A99.. Introduction Due to its important role in mathematical economics, engineering, management science, and optimization theory, conveity o unctions and sets has been studied intensively; see [3, 5, 7, 9, 0,, 6, 8, 9 and the reerences therein. Consequently, the classical concept o conve unction has been etended and generalized in dierent directions. Most important generalizations can be ound in works that change the standard requirements or a unction to be conve, thereby introducing new notions such as being quasi-conve see [8, pseudo-conve see [, strongly conve [24, approimately conve [4, midconve see [25, h-conve [27, etc. In this article, we are dealing with a recent notion o generalized conveity, introduced by I. ISCAM in [6, where he gave the ollowing deinition o harmonically conve unctions. Deinition [6. Let I be an interval in R {0}. A unction : I R is said to be harmonically conve on I i the inequality ty + t, t + ty holds, or all, y I and t [0,. Hence, harmonically conve unctions relate the harmonic mean o two points to the arithmetic mean o the unction values at the two points. Proposition [6. Let I R\{0} be a real interval and : I R a unction. hen: I I 0, + and is conve and nondecreasing, then is harmonically conve. I I 0, + and is harmonically conve and nonincreasing, then is conve. I I, 0 and is harmonically conve and nondecreasing, then is conve. I I, 0 and is conve and nonincreasing, then is harmonically conve. For some recent results, investigations, and etensions o harmonically conve unctions interested readers are reerred to [6, 4, 5, 6, 22, 23, 28. In [6 we can ind the ollowing simple but important act. [ heorem [6. I [a, b I 0, + and we consider the unction g : b, R a deined by gt =, then is harmonically conve on [a, b i and only i g is conve t in the usual sense on [ b, a.

3 Lecturas Matemáticas, vol , pp It is easy to veriy that this result is satisied i we use the interval 0, + rather than the interval [a, b. his theorem is very important, because it tells us that the graph o the unction, [ in the interval b, a b, is located below the line segment y = ab. a b a a Moreover, i a unction is harmonically conve then it satisies the ollowing inequalities, or 3 2 : , which could be considered as a harmonically perturbed conveity. he ollowing theorem on separation o unctions can be ound in the seminal papers o BARON et.al. [2, where the authors proved the ollowing sandwich theorem. heorem 2 [2. wo real unctions and g deined on a real interval I satisy t + ty tg + tgy 2 or all, y I and t [0,, i and only i there eists a conve unction h : I R such that h g. 2. Main results In this paper we have two main results. he irst one is a sandwich theorem or harmonically conve unctions, a result that is related to the theorem on separation by conve unctions presented in [2. As a second contribution, we obtain an approimate conveity result, namely, we prove a stability result o Hyers-Ulam type or harmonically conve unctions. heorem 3. Let, g be real unctions deined on the interval 0, +. he ollowing conditions are equivalent: i there eists a harmonically conve unction h : 0, + R such that h g, or all 0, +. ii the ollowing inequalities hold: tg y + tg, 3 t + ty or all, y 0, + and t [0,. Proo. [i ii We assume that there is a harmonically conve unction h : 0, + R such that h g, or all 0, +.

4 8 Mireya Bracamonte et al. A sandwich theorem and stability result o Hyers-Ulam... We consider the unctions F, G, H : 0, + R deined by F :=, G := g and H := h. Note that, by heorem, H is a unction that is conve on 0, + and satisies the inequality F H G or all 0, +. Equivalently, F u H u G u, or all u 0, +. hen, by heorem 2, the unctions F and G deined on 0, + satisy F tu + tv tgu + tgv, or all u, v 0, +, t [0,. For, y 0, + and t [0,, t + ty = F t y + t tg + tg y = tg y + tg. [ii i Conversely, i the inequalities 3 hold or all, y 0, + and t [0,, we consider the unctions F := and G := g. For all, y 0, + Equivalently, t + ty F t y + t tg y + tg tg + tg. y F tv + tu tg v + tg u, or all u, v 0, +. By heorem 2, there is a conve unction H : 0, + R such that F H G. Now, i 0, +,

5 Lecturas Matemáticas, vol , pp F where h : 0, + R, h := H o heorem. H G h g, As a consequence o heorem 3 we have the ollowing.. Note that the unction h is conve, by virtue Corollary 4. I, g, g 2 are real unctions deined on the interval 0, + and satisy the inequality tg y + tg 2 4 t + ty or all, y 0, + and t [0,, then there eists a harmonically conve unction h : 0, + R such that or all 0, +. h ma{g, g 2 } 5 Note that the reciprocal o this corollary does not hold. his can be veriied easily by making use o the unctions h = ln 3, = +, g = 2 and g 2 = 5 3. Note that these unctions satisy the inequality 5, the particular values 9 t =, = 2 and y = 3 do not satisy the inequality 4. 2 Lemma. I is a harmonically conve unction, then the unction = k + ɛ is also harmonically conve, or any constants ɛ and k R +. Proo. In act, t + ty = k t + ty + ɛ kty + t + ɛ = kty + tɛ + tk + tɛ = tky + ɛ + tk + ɛ = ty + t. he net theorem, heorem 6, is the second main result o this work, and it is about approimate conveity.

6 0 Mireya Bracamonte et al. A sandwich theorem and stability result o Hyers-Ulam... he Hyers-Ulam kind o stability problems o unctional equations was originated by ULAM in 940, when he proposed the ollowing question [26: Let be a mapping rom a group G to a metric group G 2 with metric d, such that d, y ɛ,, y G. Does there eist a group homomorphism h and δ ɛ > 0 such that d, h δ ɛ, G?. One o the irst assertions to be obtained is the ollowing result, essentially due to HYERS [2, which gives an answer to ULAM s question. heorem 5. Suppose that S is an additive semigroup, Y is a Banach space, ɛ 0, and : S Y satisies the inequality + y y ɛ, or all, y S. 6 hen there eists a unique unction A : S Y satisying A + y = A + Ay and or which A ɛ or all S. Since then, stability problems have been investigated in various directions or many other unctional equations [7. he investigation o approimate conveity probably started with the paper by HYERS and ULAM [3, who in 952 introduced and investigated ɛ-conve unctions: i D is a conve subset o a real linear space X and ɛ is a nonnegative number, then a unction : D R is called ɛ-conve i t + ty t + ty + ɛ,, y D, t [0,. HYERS and ULAM [3 proved that any ɛ conve unction where ɛ is a nonnegative number on a inite dimensional conve set can be approimated by a conve unction. As an immediate consequence o heorem 3 we obtain the ollowing stability result o Hyers-Ulam type or harmonically conve unctions see [20, 2. heorem 6. Let [a, b 0, + be an interval and ɛ > 0. A unction : [a, b R satisies the inequality ty t t + ty ɛ, 7 or all, y [a, b and t [0, i there eists an harmonically conve unctions : [a, b R such that ɛ 2, [a, b.

7 Lecturas Matemáticas, vol , pp. 5-8 Proo. Deine the unction g : [a, b R by g := + ɛ then heorem 3 holds with g = + ɛ, and it ollows that there eists a harmonically conve unction h : [a, b R such that or [a, b. h + ɛ, Putting : [a, b R deined by := h ɛ, we obtain a harmonically conve 2 unction such that ɛ 2 h ɛ 2 + ɛ 2 ɛ 2 ɛ 2 ɛ 2, or all [a, b. We will now need the ollowing setting: given > 0 and : 0, + R, we deine the unction : 0, + R by :=. he net theorem ollows rom the application o heorem 3 concerning the solutions o the inequality ty + t. 8 t + ty heorem 7. Let be a positive real number. A unction : 0, + R satisies 8 or all, y 0, + and t [0, i and only i eist a harmonically conve unction : I 0, + R such that. 9 Proo. Assume that : 0, + R satisies 8 or any, y 0, + and t [0,. We can choose λ [0, such that t = λ. Substituting λ or t in 8 we have λy + λ λ + λ y λy + λ λ + λy λy + λ, 0 λ + λy or all, y 0, + and λ [0,.

8 2 Mireya Bracamonte et al. A sandwich theorem and stability result o Hyers-Ulam... By heorem 3, there eists a harmonically conve unction h : 0, + R such that h. Deine now : 0, + R by = h. Note that = h t + ty t + ty y = h t + t y t h y + t h = ty + t. hat is, is a harmonically conve unction, and moreover = = h = h = = h =. Conversely, i there eists a harmonically conve unction : I R that satisies the inequality 9, we deine now = h. hen, h = t + ty t + ty y = t + t y y t + t = t h y = th y + th. + t h hat is, h is a harmonically conve unction. On the other hand, we have = = h = h = h = = hen, by heorem 3 we have that t + ty ty + t,

9 Lecturas Matemáticas, vol , pp or all, y 0, + and t [0,. But this means that, or all, y 0, + and t [0,, we get 8. Let I denote the real interval either 0, or [a, b R {0}, with a < b. Let : I R be a unction. Using heorem 3, we describe also solutions o the inequality t + t y + t + ty + z 0. 2 z 0 Fi a real interval I and a point z 0 I. For 0, put { I := I : z 0 z 0 I Given a real unction with the domain containing I, we deine : I R by = [ z 0 z 0. z 0 + }. Note that z 0 = [ z 2 0 z 0 + z 0 z 0 = [z 0 z 0 + z 0 = z 0. Lemma 2. I h is a harmonically conve unction, then the unction g : I R deined z by 0 g := h is harmonically conve. z 0 Proo. Let, y I and t [0,, then z 0 ty + t g = h ty + t z 0 ty + t z 0 = h z 0ty + t z 0 = h tyz 0 t + tz 0 t 2 z 2 = h 0 z 0{tyz 0 + tz 0 y} 2 z 2 0 = h z 0 yz 0 tz 0yz 0 + t z 0z 0 y z 0 yz 0

10 4 Mireya Bracamonte et al. A sandwich theorem and stability result o Hyers-Ulam... = h t th z 0y z 0 y z 0 z 0 z 0y + t z 0 y z 0 z 0 = tg + tgy. + th z 0 z 0 z 0y z 0 y Lemma 3. I satisies the inequality λy + λ λ + λy, or all, y I and λ [0,, then satisies 2 or all, y I and λ [0,. Proo. Let, y I and t [0,, we obtain ty + t ty + t z0 z0 z 0 + ty + t [ t + t y + z 0 t + t y + z 0 t + ty t + ty [t + ty + z0 t + ty + z0 hus, λ + λ y + z 0 λ + λy + z 0 or all, y I and λ [0,. heorem 8. Let 0,. A unction : I R satisies 2 or all, y I and t [0, i only i there eists a harmonically conve unction : I R such that, or I and, or I. 3 Proo. Assume that satisies 2 or any, y I and t [0,. We can choose λ [0, such that t = λ. Putting λ in place o t in 2 we get

11 Lecturas Matemáticas, vol , pp hus, λy + λ z0 z 0 λ + λy. z 0 + λy + λ or all, y I and λ [0,. λy + λ λ + λy, 4 Applying heorem 3, we obtain a harmonically conve unction h : I R such that h 5 or all I. Since z0 = z0,we have hz 0 = z 0. Deine : I := h z 0 z 0 R by + z 0. 6 By lemmas and 2, we get that is a harmonically conve unction. In addition, we will have that z 0 = z 0 and = [ z 0 z 0 z 0 + z 0 = z h 0 z 0 + z 0 + z0 z0 z 0 z 0 + z 2 0 = h z 0 + z z0 z0 z 0 + = h, I. On the other hand, or all I, we have = h z 0 z 0 + z 0 z 0 + z 0 z 0 z 0 = z 0 z0 z 0 z0 + z0 z 0 + z 0 z 2 0 z 0 = z 2 0 z0 + z0 =. z 0 Conversely, i 3 holds with a harmonically conve unction : I R then z 0 = z 0, and 6 deines a harmonically conve unction h : I R which

12 6 Mireya Bracamonte et al. A sandwich theorem and stability result o Hyers-Ulam... satisies h = hus, or any I, [ = z 0 z 0 + [ [ = h = [ z 0 z 0 + z 0 z 0 + =. z 0 z 0 + z 0. 7 z 0 z 0 z 0 By heorem 3 we obtain 4 or all, y I and t [0,. By Lemma 3, satisies 2 or all, y I and t [0,. 3. Comments In this paper we have two main results: a sandwich theorem or harmonically conve unction and an approimate conveity result, namely, we proved a stability result o Hyers- Ulam type or harmonically conve unctions. We epect that the ideas and techniques used in this paper may inspire interested readers to eplore some new applications o these unctions in various ields o pure and applied sciences. Acknowledgments he authors wish to thank the anonymous reeree or a thorough review and insightul suggestions. Reerences [ M. Avriel, W.. Diewert, S. Schaible & I. Zang, Generalized concavity, Classics in Applied Mathematics, 998. [2 K. Baron, J. Matkowski & K. Nikodem, A sandwich with conveity. Math. Pannica 5/ 994, no., [3 M. Bessenyei & Zs. Páles, Characterization o conveity via Hadamard s inequality, Math. Inequal. Appl , [4 A. Daniilidis & P. Georgiev, Approimately conve unctions and approimately monotonic operators, Nonlin. Anal , [5 S. Dragomir, Inequalities o Hermite-Hadamard type or h-conve unctions on linear spaces, Mathematica Moravica 205, [6 S. Dragomir, Inequalities o Jensen type or HA-conve unctions, RGMIA Monographs, Victoria University, 205.

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14 8 Mireya Bracamonte et al. A sandwich theorem and stability result o Hyers-Ulam... [26 S.M. Ulam. A Collection o mathematical problems, Interscience Publ., New York, 960. [27 S. Varošanec, On h-conveity, J. Math. Anal. Appl , [28.-Y Zhang, A.-P. Ji & F. Qi, Integral inequalities o Hermite-Hadamard type or harmonically quasiconve unctions, Proc. Jangjeon Math. Soc , no. 3, Received in October 206. Accepted or publication in April 207. MIREYA R. BRACAMONE P. DEPARAMENO DE MAEMÁICA, DECANAO DE CIENCIAS Y ECNOLOGÍA UNIVERSIDAD CENROCCIDENAL LISANDRO ALVARADO BARQUISIMEO-VENEZUELA FACULAD DE CIENCIAS NAURALES Y MAEMÁICA, DEPARAMENO DE MAEMÁICA ESCUELA SUPERIOR POLIÉCNICA DEL LIORAL ESPOL GUAYAQUIL-ECUADOR mireyabracamonte@ucla.edu.ve JOSÉ P. GIMÉNEZ Q. DEPARAMENO DE MAEMÁICA, FACULADA DE CIENCIAS UNIVERSIDAD DE LOS ANDES MÉRIDA - VENEZUELA jgimenez@ula.ve JESÚS G. MEDINA DEPARAMENO DE MAEMÁICA, DECANAO DE CIENCIAS Y ECNOLOGÍA UNIVERSIDAD CENROCCIDENAL LISANDRO ALVARADO BARQUISIMEO-VENEZUELA jesus.medina@ucla.edu.ve MIGUEL J. VIVAS C. FACULAD DE CIENCIAS NAURALES Y MAEMÁICA, DEPARAMENO DE MAEMÁICA ESCUELA SUPERIOR POLIÉCNICA DEL LIORAL ESPOL GUAYAQUIL-ECUADOR DEPARAMENO DE MAEMÁICA, DECANAO DE CIENCIAS Y ECNOLOGÍA UNIVERSIDAD CENROCCIDENAL LISANDRO ALVARADO BARQUISIMEO-VENEZUELA mjvivas@espol.edu.ec