Relative Strongly h-convex Functions and Integral Inequalities

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1 Appl. Mth. Inf. Sci. Lett. 4, No., (6) 39 Applied Mthemtics & Informtion Sciences Letters An Interntionl Journl Reltive Strongly h-convex Functions nd Integrl Inequlities Miguel Vivs Cortez Deprtmento de Mtemátics, Universidd Centroccidentl Lisndro Alvrdo, Brquisimeto, Venezuel Received: 4 Dec. 5, Revised: Jn. 6, Accepted: 5 Jn. 6 Published online: My 6 Abstrct: In this work we introduce the clss of functions reltive strongly h-convex functions nd we show inequlities of Hermite- Hdmrd- Fejér type. Keywords: h-convex functions, Hermite-Hdmrd, Fejér Introduction It is well known [] tht modern nlysis directly or indirectly involve the pplictions of convexity. Severl generliztions hve been introduced in recent yers nd extensions of the clssicl notion of convex function nd in the theory of inequlities re produced importnt contributions in this regrd. This reserch dels with some inequlities relted to the renowned works, on clssicl convexity, of Chrles Hermite [5], Jques Hdmrd [4] nd Lipót Fejér [3]. The inequlities of Hermite-Hdmrd nd Fejér hve been object of intense investigtion nd hve produced mny pplictions. In this pper we estblish the notion of reltive strongly h-convex function, properties nd some results relted with these inequlities mentioned bove. The Hermite-Hdmrd inequlity gives us estimte of the (integrl) men vlue of convex function; more precisely: Theorem ([4]). Let f be convex function on[, b], with <b. Then ( ) +b f b f ()+ f (b) f (x)dx. () (b ) In [3], Fejér estblished the following Fejér inequlity which is the weighted generliztion of Hermite-Hdmrd inequlity () s follows: Theorem.Let f : I R R be convex function on n intervl I nd let,b I with <b. Then ( ) +b b b f p(x)dx f()+ f(b) b p(x)dx, () where p : [, b] R is non negtive, integrble nd symmetric with respect to(+b)/. Preliminries In [] Noor introduced nd studied new clss of convex set nd convex function with respect to n rbitrry function; which re clled reltive convex set nd reltive convex function respectively, s follows. Let K be nonempty closed set in rel Hilbert spces H. Definition ([]). Let Kg be ny set in H. The set Kg is sid to be reltive convex (g-convex) with respect to n rbitrry function g : H H such tht ( t)u+ tg(v) Kg, u,v H : u,g(v) Kg, t [,]. Note tht every convex set is reltive convex, but the converse is not true. Definition ([]). A function f : Kg H is sid to be reltive convex, if there exists n rbitrry function g : H H such tht Corresponding uthor e-mil: mvivs@ucl.edu.ve c 6 NSP Nturl Sciences Publishing Cor.

2 4 M. J. Vivs Cortez: Reltive strongly h-convexity f(( t)u+ tg(v)) ( t) f(u)+ t f(g(v)) for ll u,v H : u,g(v) Kg nd t [,]. Clerly every convex function is reltive convex, but the converse is not true. The reder interested in the reltive convex functions cn consult the references [9, ]. In [] Noor estblished some Hdmrd s type inequlity for reltive convex functions s follows: Theorem 3([]). Let f : Kg=[,g(b)] R be reltive convex function. Then, we hve ( ) +g(b) f f (x)dx (g(b) ) f ()+ f (g(b)). Noor in [8] introduced the clss of reltive h-convex functions nd lso discussed some specil cses, in ddition estblished some Hermite-Hdmrd type inequlities relted to reltive h-convex functions. Definition 3([8]). A function f : Kg H is sid to be reltive h-convex function with respect to two functions h : [,] (,+ ) nd g : H H such tht Kg is reltive convex set, if f(( t)u+tg(v)) h( t) f(u)+h(t) f(g(v)) u,v H : u,g(v) Kg, t (,). Theorem 4([8]). Let f : Kg R be reltive h-convex function, such tht h( ), then, we obtin ( ) +g(b) h( ) f f (x)dx (g(b) ) [ f ()+ f (g(b))] h(t)dt. Strongly convex functions hve been introduced by Polyk in [3]. Since strong convexity is strengthening of the notion of convexity, some properties of strongly convex functions re just stronger versions of known properties of convex functions. Strongly convex functions hve been used for proving the convergence of grdient type lgorithm for minimizing function. These functions ply n importnt role in optimiztion theory nd mthemticl economics ([7, 4]). In [] H. Angulo, J. Giménez, A. Moros nd K. Nikodem estblished some Hdmrd s Type inequlity for strongly h-convex functions, this result generlizes the Hermite-Hdmrd-type inequlities obtined by N. Merentes nd K. Nikodem in [6] for strongly convex functions, s follows: Definition 4.Let (X, ) be rel normed spce, D stnds for convex subset of X, h : (,) (, ) is given function nd c is positive constnt. We sy tht function f : D R is strongly h-convex with module c if f(tx+( t)y)h(t) f(x)+h( t) f(y) ct( t) x y (3) for ll x,y D nd t (,). Theorem 5.Let h :(,) (, ) be given function. If function f : I R R is Lebesgue integrble nd strongly h-convex with module c >, then h( ) b [ f b ( +b ( f()+ f(b)) for ll,b I, <b. 3 Min results ) + c (b ) ] h(t)dt c 6 (b ), (4) In this section, we present the clss of reltive strongly h-convex functions nd discuss some importnt properties, in ddition discuss some Hermite-Hdmrd-Fejér type inequlities relted to reltive strongly h-convex functions. Definition 5.A function f : Kg H is sid to be reltive strongly h-convex function with module c > with respect to two functions h : [,] (,+ ) nd g : H H such tht Kg is reltive convex set, if f(( t)u+tg(v)) h( t) f(u)+h(t) f(g(v)) ct( t) u g(v), (5) u,v H : u,g(v) Kg, t (,). Remrk..If we tke h(t) = t in (5), then we hve the definition of reltive strongly convex function with module c..if we tke h(t) = t s in (5), then the definition of reltive strongly h-convex function with module c reduces to the definition of reltive strongly s-convex function with module c. 3.If we tke h(t) = t in (5), then the definition of reltive strongly h-convex function with module c reduces to the definition of reltive strongly Godunov-Levin function with module c. 4.If we tke h(t)= in (5), then we hve the definition of reltive strongly P-convex function with module c. 5.If we tke g(x)=x in (5), then we hve the definition of strongly h-convex function. 6.If we tke g(x) = x, h(t) = t in (5), then we hve the definition of strongly convex function with module c. c 6 NSP Nturl Sciences Publishing Cor.

3 Appl. Mth. Inf. Sci. Lett. 4, No., (6) / 4 We will present some properties for the clss of reltive strongly h-convex function. Theorem 6.Let h i : [,] (,+ ), i =, be ny two functions, α. If f i : Kg H, is reltive strongly h i - convex function with module c i >, then ()f + f is strongly h-convex function with module c> where h=mx{h,h } y c=c + c. (b)α f is reltive strongly h -convex function with module c where c=αc. Proof.(). Since ech f i : Kg H is reltive strongly h i - convex function with module c i, then u,v H : u,g(v) Kg nd t (,) we hve ( f + f )(( t)u+tg(v)) = f (( t)u+tg(v))+ f (( t)u+tg(v)) h ( t) f(u)+h (t) f(g(v)) c t( t) u g(v) +h ( t) f(u)+h (t) f(g(v)) c t( t) u g(v) h( t)( f + f )(u)+h(t)( f + f )(g(v)) (c + c )(t( t)) u g(v) h( t)( f + f )(u)+h(t)( f + f )(g(v)) c(t( t)) u g(v) where h=mx{h,h } nd c=c + c. (b). Let α R. As f : Kg H is reltive strongly h -convex function with module c u,v H : u,g(v) Kg nd t (,) we hve (α f )(( t)u+tg(v)) = α f (( t)u+tg(v)) α (h ( t) f(u)+h (t) f(g(v)) c t( t) u g(v) ) h ( t)α f(u)+h (t)α f(g(v)) αc t( t) u g(v). Therefore α f is reltive strongly h -convex function with module c where c=αc. Proposition.If f : Kg H, is reltive strongly convex function with module c> nd h :[,] (,+ ), h(t) t, then f is reltive strongly h-convex function with module c. Proof.Given tht f is reltive strongly convex function then u,v Kg : u,g(v) Kg nd t (,) we hve f(( t)u+tg(v)) ( t) f(u)+t f(g(v)) ct( t) u g(v) h( t) f(u)+h(t) f(g(v)) ct( t) u g(v). Therefore f is reltive strongly h-convex function with module c. Proposition.If f : Kg H, is reltive strongly h-convex function with module c nd h :[,] (,+ ), h(t) t, then f is reltive strongly convex function with module c>. Proof.Since f is reltive strongly h-convex function with module c then u,v Kg : u,g(v) Kg nd t (,) we hve f(( t)u+tg(v)) h( t) f(u)+h(t) f(g(v)) ct( t) u g(v) ( t) f(u)+t f(g(v)) ct( t) u g(v). Therefore f is reltive strongly convex function with module c>. Proposition 3.Let h i : [,] (,+ ), i =, be ny function such tht h (t) h (t) for t [,]. If f : Kg H is reltive strongly h -convex function with module c then f is reltive strongly h -convex function with module c with <c c. Proof.Given tht f is reltive strongly h -convex function with module c then u,v Kg : u,g(v) Kg nd t (,) we hve f(( t)u+tg(v)) h ( t) f(u)+h (t) f(g(v)) c t( t) u g(v) h ( t) f(u)+h (t) f(g(v)) ct( t) u g(v). Therefore f is reltive strongly h -convex function with module c with <c c. Proposition 4.If f n : Kg H, is sequence of functions which pointwise converge to f : Kg H nd h n : [,] (,+ ), is sequence of functions which pointwise converge to h : [,] (,+ ) so there is k > such tht f n is reltive strongly h n -convex function with module c n for n k, then f is reltive strongly h-convex function with module c, where c = lim c n. n + Proof.As ech f n is reltive strongly h n -convex function with module c n then u,v Kg : u,g(v) Kg nd t (,) we hve f(( t)u+tg(v)) = lim f n(( t)u+tg(v)) n + lim (h n ( t) f n (u)+h n (t) f n (g(v)) c n t( t) u g(v) ) n + h( t) f(u)+h(t) f(g(v)) ct( t) u g(v). Therefore f is reltive strongly h-convex function with module c, where c = lim n + c n. Proposition 5.Let f i : Kg H,g : H H, h i : [,] (,+ ), be with i =,. If f i is reltive strongly h i -convex function with module c i with i =, then f(x) = mx{ f (x), f (x)} is reltive strongly h-convex function with module c, where h(t)=mx{h (t),h (t)}. c 6 NSP Nturl Sciences Publishing Cor.

4 4 M. J. Vivs Cortez: Reltive strongly h-convexity Proof.Since ech f i is reltive strongly h i -convex function with module c i then u,v Kg : u,g(v) Kg nd t (,) we hve f i (( t)u+tg(v)) h i ( t) f i (u)+h i (t) f i (g(v)) c i t( t) u g(v) h( t) f(u)+h(t) f(g(v)) ct( t) u g(v). This implies tht f(( t)u+tg(v)) = mx{ f (( t)u+tg(v)), f (( t)u+tg(v))} h( t) f(u)+h(t) f(g(v)) ct( t) u g(v). Therefore f(x) = mx{ f (x), f (x)} is reltive strongly h-convex function with module c, where h(t)=mx{h (t),h (t)}. Proposition 6.Let h : [, ] (, + ) be given function. If g : H H, is invertible nd f : Kg H is reltive strongly h-convex function with module c, then f is strongly h-convex function with module c. Proof.Since f is reltive strongly h-convex function with module c then u,v Kg : u,g(v) Kg nd t (,) we get f(( t)u+tv) = f(( t)u+t(g(g (v)))) h( t) f(u)+h(t) f(g(g (v))) c t( t) u g(g (v)) h( t) f(u)+h(t) f(v) ct( t) u (v). Therefore f is strongly h-convex function with module c. Note tht the previous theorem shows us tht if g : H H, is invertible then the set of the reltive strongly h-convex functions with module c is contined in the set of the strongly h-convex functions with module c. Proposition 7.If f :[,g(b)] R is reltive strongly h- convex function with module c nd h :[,] (,+ ) is n upper bounded function then f is n upper bounded function. Proof.For ny x=( t)+ tg(b) [,g(b)] we obtin f(x) h( t) f()+h(t) f(g(b)) c t( t)( g(b)) M f()+m f(g(b)) ct( t)( g(b)) M( f()+ f(g(b))). Therefore f is n upper bounded function. Theorem 7.Let h : (,) (, ) be given function. If function f : I R is Lebesgue integrble nd reltive strongly h-convex with module c >, then g(b) ( f()+ f(g(b))) h(t)dt c 6 (g(b) ), (6) for ll,g(b) I, <g(b). Proof.Tke ( t)+ tg(b) [,g(b)]. Then, the reltive strong h-convexity of f implies f(( t)+ tg(b)) h( t) f()+h(t) f(g(b)) ct( t)( g(b)). Integrting over the intervl(, ), we get f(( t)+tg(b))dt f() c g(b) t( t)dt. ( h( t) f()+h(t) f(g(b)) ct( t)( g(b)) ) dt h( t)dt+ f(g(b)) h(t)dt By simple clcultion nd using the chnge of the vrible, we obtin g(b) ( f()+ f(g(b))) h(t)dt c 6 (g(b) ). (7) Remrk.If g : R R is right-invertible, f : K g = [,g(b)] R is reltive strongly h-convex function nd f is Lebesgue integrble then f is strongly h-convex nd we get the following result is counterprt of the Hermite-Hdmrd inequlities for reltive strongly h-convex functions. [ ( ) +g(b) ( ) f + c ] (g(b) ) h g(b) ( f()+ f(g(b))) h(t)dt c 6 (g(b) ), (8) nd when g is the identity function then the result (8) coincides with the Theorem 4. in []. 4 A refinement of the Hermite-Hdmrd type inequlities In this section we present refinement of the right-hnd side of the Hermite-Hdmrd type inequlities (8) for reltive strongly h-convex functions. A similr result for strongly convex functions cn be found in [, Theorem 5]. Theorem 8.Let h : (,) (, ) be given function. If function f : I R is Lebesgue integrble nd reltive c 6 NSP Nturl Sciences Publishing Cor.

5 Appl. Mth. Inf. Sci. Lett. 4, No., (6) / 43 strongly h-convex with module c >, then g(b) +h )) ( f()+ f(g(b))) h(t)dt [ h(t)dt+ ] c(g(b) ), (9) 4 4 for ll,g(b) I, <g(b). Proof.Applying [ the Theorem 7 in the intervls, +g(b) ] [ ] +g(b) nd,g(b) we obtin Corollry.Under the sme hypotheses of theorem 8, if h( ) nd h(t)dt we get g(b) +h )) ( f()+ f(g(b))) h(t)dt [ h(t)dt+ ] c(g(b) ) 4 4 ( f()+ f(g(b))) h(t)dt c 6 (g(b) ). +g(b) g(b) )) +g(b) f()+ f h(t)dt c (g(b) ), () 6 4 nd g(b) +g(b) g(b) ) ) +g(b) f + f(g(b)) h(t)dt c (g(b) ). () 6 4 Summing up these inequlities we get g(b) g(b) ( ) +g(b) ( f()+f + f(g(b))) h(t)dt c (g(b) ). 6 4 Therefore g(b) ) ) +g(b) f()+f + f(g(b)) h(t)dt c (g(b) ). 6 4 Now, using the reltive strong h-convexity of f, we obtin ( ) +g(b) f ( ) ( ) h f()+h f(g(b)) c 4 ( g(b)). Thus, g(b) +h )) ( f()+ f(g(b))) h(t)dt c 4 ( g(b)) h(t)dt c (g(b) ) 6 4 +h )) = ( f()+ f(g(b))) h(t)dt [ h(t)dt+ ] c(g(b) ). 4 4 Corollry.If we tke g(b) = b, then we get the right-hnd side of the inequlity given in []. Remrk..If we tke c= nd h( )= in the Theorem 9, then we hve the right-hnd side of the inequlity given in [8, Theorem 6]..If we tke h(t)= t s with s [,] in Corollry, then we obtin t s dt = s+ s, nd ( ) h s s s thus, the theorem is vlid only for s=. 3.If we tke h(t)= t for t (,) then the inequlities in the Corollry reduce to ( f()+ f(g(b))) c g(b) 6 (g(b) ), these is the hermite-hdmrd type inequlities for reltive strongly convex functions. 5 Fejér type inequlities Now we will present bound for the right hnd side of (). First, we prove the following result which is similr to Lemm in [5]. Lemm.If f :[, ) R is reltive strongly h-convex function, with module c >, then, for ll x [,g(b)] [, ) there is α x [,] such tht f(+g(b) x) h( α x ) f()+h(α x ) f(g(b)) c(x )(g(b) x). () where α x = x g(b) nd α x = g(b) x g(b). c 6 NSP Nturl Sciences Publishing Cor.

6 44 M. J. Vivs Cortez: Reltive strongly h-convexity Proof.Since ny x [, g(b)] cn be written s x=α x +( α x )g(b), for some α x [,], where α x = α x = g(b) x g(b) nd x g(b), +g(b) x= +g(b) α x ( α x )g(b)=( α x )+α x g(b), we get f(+g(b) x) = f(+g(b) α x ( α x )g(b)) = f(( α x )+α x g(b)) h( α x ) f()+h(α x ) f(g(b)) c(x )(g(b) x). The proof is completed. Theorem 9.Let f : [, ) R be reltive strongly h-convex function with module c >, which is integrble in [,g(b)], where,g(b) [, ), < g(b), nd let p : [, g(b)] R be non negtive nd integrble function which is symmetric with respect to +g(b), then g(b) g(b) [h( α x ) f()+h(α x ) f(g(b)) c(x )(g(b) x)]p(x)dx. where α x = x g(b) nd α x = g(b) x g(b). Proof.By the symmetry of p with respect to +g(b) nd Lemm g(b) = g(b) f(+g(b) x)p(+g(b) x)dx+ g(b) = g(b) f(+g(b) x)p(x)dx+ g(b) g(b) [h( α x ) f()+h(α x ) f(g(b)) c(x )(g(b) x)]p(x)dx + g(b) f() g(b) h( α x )p(x)dx+ f(g(b)) g(b) h(α x )p(x)dx g(b) c(x )(g(b) x)p(x)dx+, thus g(b) = g(b) f(+g(b) x)p(+g(b) x)dx + g(b) f() g(b) g(b) c(x )(g(b) x)p(x)dx. h( α x )p(x)dx+ f(g(b)) g(b) h(α x )p(x)dx Remrk.Notice tht if h(t) = in Theorem 9 we, indeed, get g(b) ( ) f()+ f(g(b)) g(b) p(x)dx g(b) c(x )(g(b) x)p(x)dx for reltive strongly P-convex functions with module c. We expect tht the ides nd techniques used in this pper my inspire interested reders to explore some new pplictions of these newly introduced functions in vrious fields of pure nd pplied sciences. Acknowledgements We wnt to give thnks to the librry stff of Centrl Bnk of Venezuel (BCV) for compiling the references. References [] H. Angulo, J. Giménez, A, Moros, K. Nikodem; on strongly h-convex functions.ann. Funct. Anl.(),85-9 (). [] A. Azocr, K. Nikodem nd G. Ro, Fejér-Type inequlities for strongly convex function, Annles Mthemtice Silesine 6 (), 4354 [3] L. Fejér. Uber die Fourierreinhen,II. Mth. Nturwiss. Anz. Ungr. Akdd. Wiss. 4 (96) [4] J.S Hdmrd. Etude sur les propiètés des fonctions entieres et en prticulier d une fontion considerer per Riemnn, J. Mth. Pure nd Appl. 58 (893) 7-5 [5] Ch. Hermite, Sur deux limites d une intégrle défine, Mthesis 3, (883), 8. [6] N. Merentes nd K. Nikodem, Remrks on strongly convex functions, Aequtiones Mth. 8 (), no. -, [7] L. Montrucchio,Lipschitz continuous policy functions for strongly concve optimiztion problems, J. Mth. Econ.,6(987), [8] M. A. Noor, K. I. Noor nd M. U. Awn, Generlized convexity nd integrl inequlities, Appl. Mth. Inf. Sci.9, No., (5). [9] M. A. Noor: Advnced convex nlysis, Lecture Notes, Mthemtics Deprtment, COMSATS Institute of Informtion Technology, Islmbd, Pkistn,. [] M. A. Noor: On some chrcteriztions of nonconvex functions, Nonliner Anlysis Forum, 93, (7). [] M. A. Noor: Differentible non-convex functions nd generl vritionl inequlities, Appl. Mth. Comp.99,6363, (8) [] J. E. Pečrić, F. Proschn, Y. L. Tong, Convex Functions, Prtil Orderings, nd Sttisticl Appli- ctions, Acd. Press,Inc., Boston, 99. [3] B. T. Polyk, Existence theorems nd convergence of minimizing sequence sin extremum problems with restrictions, Soviet Mth. Dokl. 7 (966), c 6 NSP Nturl Sciences Publishing Cor.

7 Appl. Mth. Inf. Sci. Lett. 4, No., (6) / 45 [4] A. W. Roberts nd D. E. Vrberg,Convex functions. Acdemic Pres. New York [5] M. Z. Sriky, E. Set nd M. E. Ozdemir,On some new inequlities of Hdmrd type involving h-convex functions, Act Mth. Univ Comenine, Vol LXXIX, (), Miguel Vivs Cortez erned his PhD degree from Universidd Centrl de Venezuel, Crcs, Distrito Cpitl (4) in the field Pure Mthemtics (Nonliner Anlysis). He hs vst experience of teching nd reserch t university levels. It covers mny res of Mthemticl such s Inequlities, Bounded Vrition Functions nd Ordinry Differentil Equtions. He hs written nd published severl reserch rticles in reputed interntionl journls of mthemticl nd textbooks. He is currently Professor in Decnto de Ciencis y Tecnologi of Universidd Centroccidentl Lisndro Alvrdo (UCLA), Brquisimeto, Lr stte. He hs been involved with the reserch group University Centrl of Venezuel - Bnco Centrl de Venezuel (UCV-BCV). c 6 NSP Nturl Sciences Publishing Cor.