INEQUALITIES OF HERMITE-HADAMARD TYPE FOR

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1 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE FOR COMPOSITE h-convex FUNCTIONS SILVESTRU SEVER DRAGOMIR ; Abstrct. In this pper we obtin some inequlities of Hermite-Hdmrd type for composite conve functions. Applictions for AG, AH-h-conve functions, GA; GG; GH-h-conve functions nd HA; HG; HH-h-conve function re given.. Introduction We recll here some concepts of conveity tht re well known in the literture. Let I be n intervl in R. De nition ([5]). We sy tht f : I! R is Godunov-Levin function or tht f belongs to the clss Q (I) if f is non-negtive nd for ll ; y I nd t (; ) we hve (.) f (t + ( t) y) t f () + t f (y) : Some further properties of this clss of functions cn be found in [4], [43], [45], [58], [64] nd [65]. Among others, its hs been noted tht non-negtive monotone nd non-negtive conve functions belong to this clss of functions. De nition ([45]). We sy tht function f : I! R belongs to the clss P (I) if it is nonnegtive nd for ll ; y I nd t [; ] we hve (.) f (t + ( t) y) f () + f (y) : Obviously Q (I) contins P (I) nd for pplictions it is importnt to note tht lso P (I) contin ll nonnegtive monotone, conve nd qusi conve functions, i. e. nonnegtive functions stisfying (.3) f (t + ( t) y) m ff () ; f (y)g for ll ; y I nd t [; ] : For some results on P -functions see [45] nd [6] while for qusi conve functions, the reder cn consult [44]. De nition 3 ([]). Let s be rel number, s (; ]: A function f : [; )! [; ) is sid to be s-conve (in the second sense) or Breckner s-conve if for ll ; y [; ) nd t [; ] : f (t + ( t) y) t s f () + ( t) s f (y) 99 Mthemtics Subject Clssi ction. 6D5; 6D. Key words nd phrses. Conve functions, AG, AH-h-conve functions, GA; GG; GH-hconve functions nd HA; HG; HH-h-conve function, Integrl inequlities. 8 by the uthor(s). Distributed under Cretive Commons CC BY license.

2 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v S. S. DRAGOMIR For some properties of this clss of functions see [], [3], [], [], [4], [4], [53], [55] nd [67]. In order to unify the bove concepts for functions of rel vrible, S. Vrošnec introduced the concept of h-conve functions s follows. Assume tht I nd J re intervls in R; (; ) J nd functions h nd f re rel non-negtive functions de ned in J nd I; respectively. De nition 4 ([7]). Let h : J! [; ) with h not identicl to. We sy tht f : I! [; ) is n h-conve function if for ll ; y I we hve (.4) f (t + ( t) y) h (t) f () + h ( t) f (y) for ll t (; ) : For some results concerning this clss of functions see [7], [9], [56], [68], [66] nd [69]. We cn introduce now nother clss of functions. De nition 5. We sy tht the function f : I! [; ) is of s-godunov-levin type, with s [; ] ; if (.5) f (t + ( t) y) t s f () + ( t) s f (y) ; for ll t (; ) nd ; y I: We observe tht for s = we obtin the clss of P -functions while for s = we obtin the clss of Godunov-Levin. If we denote by Q s (C) the clss of s- Godunov-Levin functions de ned on C, then we obviously hve P (C) = Q (C) Q s (C) Q s (C) Q (C) = Q (C) for s s : If f : I! [; ) is n h-conve function on n intervl I of rel numbers with h L [; ] nd f L [; b] with ; b I; < b; then we hve the Hermite-Hdmrd type inequlity obtined by Sriky et l. in [66] + b f b (.6) f (u) du b h = f (( ) + b) d [f () + f (b)] : For n etension of this result to functions de ned on conve subsets of liner spces nd re nements, see [3]. In order to etend this result for other clsses of functions, we need the following preprtions. Let g : [; b]! [g () ; g (b)] be continuous strictly incresing function tht is di erentible on (; b) : De nition 6. A function f : [; b]! R will be clled composite-g h-conve (concve) on [; b] if the composite function f g : [g () ; g (b)]! R is h-conve (concve) in the usul sense on [g () ; g (b)] : If f : [; b]! R is composite-g h-conve on [; b] then we hve the inequlity (.7) f g (( ) u + v) h ( ) f g (u) + f g (v) for ny u; v [g () ; g (b)] nd [; ] :

3 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 3 This is equivlent to the condition (.8) f g (( ) g (t) + g (s)) h ( ) f (t) + f (s) for ny t; s [; b] nd [; ] : If we tke g (t) = ln t, t [; b] (; ) ; then the condition (.8) becomes (.9) f t s h ( ) f (t) + f (s) for ny t; s [; b] nd [; ] ; which is the concept of GA-h-conveity s considered in []. If we tke g (t) = t ; t [; b] (; ) ; then (.8) becomes ts (.) f h ( ) f (t) + f (s) ( ) s + t for ny t; s [; b] nd [; ] ; which is the concept of HA-h-conveity s considered in [5]. If p > nd we consider g (t) = t p ; t [; b] (; ) ; then the condition (.8) becomes h (.) f (( ) t p + s p ) =pi h ( ) f (t) + f (s) for ny t; s [; b] nd [; ] : For h (t) = t the concept of p-conveity ws considered in [7]. If we tke g (t) = ep t; t [; b] ; then the condition (.8) becomes (.) f [ln (( ) ep (t) + ep g (s))] ( ) f (t) + f (s) ; which is the concept of LogEp h-conve function on [; b] : For h (t) = t; the concept ws considered in [8]. Further, ssume tht f : [; b]! J ; J n intervl of rel numbers nd k : J! R continuous function on J tht is strictly incresing (decresing) on J : De nition 7. We sy tht the function f : [; b]! J is k-composite h-conve (concve) on [; b], if k f is h-conve (concve) on [; b] : With g : [; b]! [g () ; g (b)] continuous strictly incresing function tht is di erentible on (; b) ; f : [; b]! J ; J n intervl of rel numbers nd k : J! R continuous function on J tht is strictly incresing (decresing) on J ; we cn lso consider the following concept: De nition 8. We sy tht the function f : [; b]! J is k-composite-g h-conve (concve) on [; b] ; if k f g is h-conve (concve) on [g () ; g (b)] : This de nition is equivlent to the condition (.3) k f g (( ) g (t) + g (s)) h ( ) (k f) (t) + (k f) (s) for ny t; s [; b] nd [; ] : If k : J! R is strictly incresing (decresing) on J ; then the condition (.3) is equivlent to: (.4) f g (( ) g (t) + g (s)) for ny t; s [; b] nd [; ] : () k [h ( ) (k f) (t) + (k f) (s)]

4 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v 4 S. S. DRAGOMIR If k (t) = ln t; t > nd f : [; b]! (; ), then the fct tht f is k-composite h-conve on [; b] is equivlent to the fct tht f is log-conve or multiplictively conve or AG-h-conve, nmely, for ll ; y I nd t [; ] one hs the inequlity: (.5) f (t + ( t) y) [f ()] h(t) [f (y)] h( t) : A function f : I! Rn fg is clled AH-h-conve (concve) on the intervl I if the following inequlity holds [] (.6) f (( ) + y) () h ( f () f (y) ) f (y) + f () for ny ; y I nd [; ] : An importnt cse tht provides mny emples is tht one in which the function is ssumed to be positive for ny I: In tht sitution the inequlity (.6) is equivlent to h ( ) f () + f (y) () f (( ) + y) for ny ; y I nd [; ] : Tking into ccount this fct, we cn conclude tht the function f : I! (; ) is AH-h-conve (concve) on I if nd only if f is k-composite h-concve (conve) on I with k : (; )! (; ) ; k (t) = t : Following [], we cn introduce the concept of GH-h-conve (concve) function f : I (; )! R on n intervl of positive numbers I s stisfying the condition (.7) f y () Since nd f () f (y) h ( ) f (y) + f () : f y = f ep [( ) ln + ln y] f () f (y) h ( ) f (y) + f () = f ep (ln ) f ep (ln y) h ( ) f ep (y) + f ep () then f : I (; )! R is GH-conve (concve) on I if nd only if f ep is AHconve (concve) on ln I := fj = ln t; t Ig : This is equivlent to the fct tht f is k-composite-g h-concve (conve) on I with k : (; )! (; ) ; k (t) = t nd g (t) = ln t; t I: Following [], we sy tht the function f : I R n fg! (; ) is HH-h-conve if y f () f (y) (.8) f t + ( t) y h ( t) f (y) + h (t) f () for ll ; y I nd t [; ]. If the inequlity in (.8) is reversed, then f is sid to be HH-h-concve. We observe tht the inequlity (.8) is equivlent to (.9) h ( t) f () + h (t) f (y) f y t+( t)y for ll ; y I nd t [; ]. This is equivlent to the fct tht f is k-composite-g h-concve on [; b] with k : (; )! (; ) ; k (t) = t nd g (t) = t ; t [; b] :

5 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 5 The function f : I (; )! (; ) is clled GG-h-conve on the intervl I of rel umbers R if [5] (.) f y [f ()] h( ) [f (y)] h() for ny ; y I nd [; ] : If the inequlity is reversed in (.) then the function is clled GG-h-concve. For h (t) = t; this concept ws introduced in 98 by P. Montel [59], however, the roots of the reserch in this re cn be trced long before him [6]. It is esy to see tht [6], the function f : [; b] (; )! (; ) is GG-h-conve if nd only if the the function g : [ln ; ln b]! R, g = ln f ep is h-conve on [ln ; ln b] : This is equivlent to the fct tht f is k-composite-g h-conve on [; b] with k : (; )! R; k (t) = ln t nd g (t) = ln t; t [; b] : Following [] we sy tht the function f : I R n fg! (; ) is HG-h-conve if (.) f y t + ( [f ()] h( t) [f (y)] h(t) t) y for ll ; y I nd t [; ]. If the inequlity in (.8) is reversed, then f is sid to be HG-h-concve. Let f : [; b] (; )! (; ) nd de ne the ssocited functions G f : b ;! R de ned by G f (t) = ln f t : Then f is HG-h-conve on [; b] i Gf is h-conve on b ; : This is equivlent to the fct tht f is k-composite-g h-conve on [; b] with k : (; )! R; k (t) = ln t nd g (t) = t ; t [; b] : We sy tht the function f : [; b]! (; ) is r-h-conve, for r 6= ; if (.) f (( ) + y) [h ( ) f r (y) + f r ()] =r for ny ; y [; b] nd [; ]. For h (t) = t; the concept ws considered in [6], If r > ; then the condition (.) is equivlent to f r (( ) + y) h ( ) f r (y) + f r () nmely f is k-composite conve on [; b] where k (t) = t r ; t : If r < ; then the condition (.) is equivlent to f r (( ) + y) h ( ) f r (y) + f r () nmely f is k-composite h-concve on [; b] where k (t) = t r ; t > : In this pper we obtin some inequlities of Hermite-Hdmrd type for composite conve functions. Applictions for AG, AH-h-conve functions, GA; GG; GH-h-conve functions nd HA; HG; HH-h-conve function re given.. Refinements of HH-Inequlity The following representtion result holds. Lemm. Let f : I R! C where I is n intervl of the rel numbers R: Let y; I with y 6= nd ssume tht the mpping [; ] 3 t 7! f g [( t) g () + tg (y)] is Lebesgue integrble on [; ] : Then for ny [; ]

6 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v 6 S. S. DRAGOMIR we hve the representtion (.) f g [( = ( ) t) g () + tg (y)] dt f g [( t) (( ) g () + g (y)) + tg (y)] dt + f g [( t) g () + t (( ) g () + g (y))] dt: Proof. For = nd = the equlity (.) is obvious. Let (; ) : Observe tht nd = f g [( t) (g (y) + ( ) g ()) + tg (y)] dt = f g [(( t) + t) g (y) + ( t) ( ) g ()] dt f g [t (g (y) + ( ) g ()) + ( t) g ()] dt f g [tg (y) + ( t) g ()] dt: If we mke the chnge of vrible u := ( t) + t then we hve u = ( t) ( ) nd du = ( ) du: Then f g [(( t) + t) g (y) + ( t) ( ) g ()] dt = f g [ug (y) + ( u) g ()] du: If we mke the chnge of vrible u := t then we hve du = dt nd Therefore f g [tg (y) + ( ( ) + = = t) g ()] dt = f g [ug (y) + ( f g [( t) (g (y) + ( ) g ()) + tg (y)] dt f g [t (g (y) + ( ) g ()) + ( t) g ()] dt f g [ug (y) + ( u) g ()] du + f g [ug (y) + ( nd the identity (.) is proved. u) g ()] du f g [ug (y) + ( u) g ()] du: u) g ()] du

7 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 7 Theorem. Assume tht the function f : I R! [; ) is composite-g h-conve function with h L [; ] : Let y; I with y 6=, then for ny [; ] we hve the inequlities (.) h ( ) g () + ( + ) g (y) ( ) f g ( ) g () + g (y) +f g y f (t) g (t) dt g (y) g () f g (( ) g () + g (y)) + ( ) f (y) + f () f[h ( ) + ] f () + [ + ] f (y)g : If f : I R! [; ) is composite-g h-concve function, then the inequlities reverse in (.). Proof. Since f : I R! [; ) is composite-g h-conve function function, then by Hermite-Hdmrd type inequlity (.6) we hve (.3) h f g ( ) g () + ( + ) g (y) f g [( t) (( ) g () + g (y)) + tg (y)] dt f g (( ) g () + g (y)) + f (y) nd (.4) h f g ( ) g () + g (y) f g [( t) g () + t (( ) g () + g (y))] dt f () + f g (( ) g () + g (y)) : Now, if we multiply the inequlity (.3) by the obtined inequlities, then we get (.5) ( f g h + ( f h g nd (.4) by nd dd ) g () + ( + ) g (y) ) g () + g (y)

8 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v 8 S. S. DRAGOMIR ( ) + f g [( t) (( ) g () + g (y)) + tg (y)] dt f g [( t) g () + t (( ) g () + g (y))] dt ( ) f g (( ) g () + g (y)) + f (y) + f () + f g (( ) g () + g (y)) nd by (.) we obtin (.6) h ( ) g () + ( + ) g (y) ( ) f g +f g ( f g [( ) g () + g (y) t) g () + tg (y)] dt f g (( ) g () + g (y)) + ( ) f (y) + f () f[h ( ) + ] f () + [ + ] f (y)g ; where the lst inequlity follows by the de nition of composite-g h-conveity nd performing the required clcultion. By using the chnge of vrible u = ( t) g ()+tg (y) ; we hve du = (g (y) g ()) dt nd then f g [( t) g () + tg (y)] dt = g (y) g () g(y) g() f g (u) du: If we chnge the vrible t = g (u) ; then u = g (t) ; which gives tht du = g (t) dt nd then g(y) f g (u) du = y g() f (t) g (t) dt nd the inequlity (.) is obtined. Remrk. With the ssumptions from Theorem, we observe tht if we tke either = or = in the rst two inequlities in (.), then we get (.6).

9 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 9 If we tke = nd use the h-conveity of f g ; then we get from (.) tht g () + g (y) (.7) 4h f g g () + 3g (y) 3g () + g (y) 4h f g + f g 4 4 y f (t) g (t) dt g (y) g () g () + g (y) f g h where y; I with y 6= : + + [f () + f (y)] f () + f (y) ; Remrk. In generl, if > for (; ) ; then for y; I with y 6= ( ) g () + ( + ) g (y) ( ) f g ( ) g () + g (y) + f g = ( ) g () + ( + ) g (y) h ( ) f g h ( ) + ( ) g () + g (y) f g min h ( ) ; ( ) g () + ( + ) g (y) h ( ) f g ( ) g () + g (y) + f g min h ( ) ; f g ( ) g () + ( + ) g (y) ( ) g () + g (y) ( ) + = min h ( ) ; nd from (.) we get the sequence of inequlities (.8) min h ( ) ; h h f g g () + g (y) g () + g (y) f g ( ) f g ( ) g () + ( + ) g (y) +f g ( ) g () + g (y)

10 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v S. S. DRAGOMIR g (y) g () y f (t) g (t) dt f g (( ) g () + g (y)) + ( ) f (y) + f () f[h ( ) + ] f () + [ + ] f (y)g ; for y; I with y 6= : In prticulr, we hve (.9) 4h g () + g (y) f g ( 4h f g ) g () + ( + ) g (y) +f g ( ) g () + g (y) g (y) g () y f (t) g (t) dt f g g () + g (y) + h f (y) + f () + [f () + f (y)] : In similr wy, if f : I R! [; ) is composite-g h-concve function, then (.) h m h g () + g (y) f g h ( ) ; ( ) g () + ( + ) g (y) ( ) f g +f g ( ) g () + g (y) g (y) g () y f (t) g (t) dt f g (( ) g () + g (y)) + ( ) f (y) + f () f[h ( ) + ] f () + [ + ] f (y)g for y; I with y 6= : ;

11 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE In prticulr, (.) 4h g () + g (y) f g ( 4h f g ) g () + ( + ) g (y) +f g ( ) g () + g (y) g (y) g () y f (t) g (t) dt f g g () + g (y) + h f (y) + f () + [f () + f (y)] : Corollry. Let f : I R! [; ) be composite-g conve function on the intervl I in R: Then for ny y; I with y 6= nd for ny [; ] we hve the inequlities g () + g (y) ( ) g () + ( + ) g (y) (.) f g ( ) f g ( ) g () + g (y) + f g y f (t) g (t) dt g (y) g () f g (( ) g () + g (y)) + ( ) f (y) + f () f (y) + f () : We hve: Corollry. Let f : I R! [; ) be composite-g Breckner s-conve function on the intervl I with s (; ]. Then for ny y; I with y 6= nd for ny [; ] we hve the inequlities (.3) s s s g () + g (y) f g ( ) g () + ( + ) g (y) ( ) f g +f g ( ) g () + g (y)

12 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v S. S. DRAGOMIR y (.4) f (t) g (t) dt g (y) g () f g (( ) g () + g (y)) + ( ) f (y) + f () s + s + f[( )s + ] f () + ( s + ) f (y)g : We lso hve: Corollry 3. Let f : I R! [; ) be composite-g of s-godunov-levin type on the intervl I with s (; ). Then for ny y; I with y 6= nd for ny (; ) we hve the inequlities (.5) +s s +s More generlly, we hve: +s g () + g (y) f g ( ) g () + ( + ) g (y) ( ) f g ( ) g () + g (y) +f g y f (t) g (t) dt g (y) g () f g (( ) g () + g (y)) + ( ) f (y) + f () nh i ( ) s + f () + s + s o f (y) : Corollry 4. Assume tht g : [; b]! [g () ; g (b)] is continuous strictly incresing function tht is di erentible on (; b) ; f : [; b]! J ; J n intervl of rel numbers nd k : J! R is continuous function on J tht is strictly incresing. If the function f : [; b]! J is k-composite-g h-conve on [; b] ; then (.6) h min h ( ) ; h y k f g g () + g (y) ( ) k f g ( ) g () + ( + ) g (y) +k f g ( ) g () + g (y) (k f) (t) g (t) dt g (y) g () k f g (( ) g () + g (y)) + ( ) (k f) (y) + (k f) () f[h ( ) + ] (k f) () + [ + ] (k f) (y)g for y; [; b] with y 6= : ;

13 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 3 If the function f : [; b]! J is k-composite-g h-concve on [; b] ; then (.7) h m h g () + g (y) k f g h ( ) ; ( ) g () + ( + ) g (y) ( ) k f g +k f g ( ) g () + g (y) y (k f) (t) g (t) dt g (y) g () k f g (( ) g () + g (y)) + ( ) (k f) (y) + (k f) () f[h ( ) + ] (k f) () + [ + ] (k f) (y)g for y; [; b] with y 6= : ; The proof follows by the inequlities (.8) nd (.) nd we omit the detils. In 96, Fejér [5], while studying trigonometric polynomils, obtined the following inequlities which generlize tht of Hermite & Hdmrd: Theorem (Fejér s Inequlity). Consider the integrl R b h () w () d, where h is conve function in the intervl (; b) nd w is positive function in the sme intervl such tht w () = w ( + b ) ; for ny [; b] i.e., y = w () is symmetric curve with respect to the stright line which contins the point ( + b) ; nd is norml to the -is. Under those conditions the following inequlities re vlid: + b b (.8) h w () d b h () w () d h () + h (b) If h is concve on (; b), then the inequlities reverse in (.8). b w () d: If w : [; b]! R is continuous nd positive on the intervl [; b] ; then the function W : [; b]! [; ); W () := R h w (s) ds is strictly incresing nd di erentible on (; b) nd the inverse W : ; R i b w (s) ds! [; b] eists. Remrk 3. Assume tht w : [; b]! R is continuous nd positive on the intervl [; b], f : [; b]! J ; J n intervl of rel numbers nd k : J! R is continuous

14 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v 4 S. S. DRAGOMIR function on J tht is strictly incresing. If the function f : [; b]! J is k- composite-w h-conve on [; b] ; then we hve the weighted inequlity (.9) h h min ( h ( ) ; R k f W w (s) ds + R y " R ( ) ( ) k f W w (s) ds + ( + ) R y " R ( ) +k f W w (s) ds + R y! w (s) ds # w (s) ds #) w (s) ds y R y w (s) ds (k f) (t) w (t) dt k f W ( ) w (s) ds + y w (s) ds i + ( ) (k f) (y) + (k f) () f[h ( ) + ] (k f) () + [ + ] (k f) (y)g for ny [; ] nd for y; [; b] with y 6= : ; 3. Applictions for AG nd AH-h-Conve Functions The function f : [; b]! (; ) is AG-h-conve mens tht f is k-composite h-conve on [; b] with k (t) = ln t; t > : By mking use of Corollry 4 for g (t) = t; we get (3.) f + y h( ) minf h( ) ; h()g ( ) + ( + ) y ( f ( ) f ) + y h ( ) ep y y ln f (t) dt h i R f (( ) + y) f ( ) (y) f () h(t)dt n o R f [h( )+] () f [h()+ ] (y) h(t)dt ; for ny [; ] nd ; y [; b] with y 6= : The function f : [; b]! (; ) is AH-h-conve on [; b] mens tht f is k- composite h-concve on [; b] with k : (; )! (; ) ; k (t) = t : By mking use

15 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 5 of Corollry 4 for g (t) = t; we get (3.) h m h ( ) ; h + y f ( ) + ( + ) y ( ) f +f ( ) + y y y f (t) dt f (( ) + y) + ( ) f (y) + f () [h ( ) + ] f () + [ + ] f (y) for ny [; ] nd ; y [; b] with y 6= : ; 4. Applictions for GA; GG nd GH-h-Conve Functions If we tke g (t) = ln t, t [; b] (; ) ; then f : [; b]! R is GA-h-conve on [; b] mens tht tht f : [; b]! R composite-g h-conve on [; b] : By mking use of Corollry 4 for k (t) = t; we get (4.) h min h ( ) ; f ( p y) h n o ( ) f + y + f y y ln y f (t) dt t f y + ( ) f (y) + f () f[h ( ) + ] f () + [ + ] f (y)g ; for ny [; ] nd for y; [; b] with y 6= : The function f : I (; )! (; ) is GG-h-conve mens tht f is k- composite-g h-conve on [; b] with k : (; )! R; k (t) = ln t nd g (t) = ln t; t [; b] : By mking use of Corollry 4 we get (4.) [f ( p y)] h( ) minf h( ) ; h()g nf o ( ) + y f y h( )

16 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v 6 S. S. DRAGOMIR ep y ln y! ln f (t) dt t f y f () f (y) R h(t)dt n o R f [h( )+] () f [h()+ ] (y) h(t)dt ; for ny [; ] nd for y; [; b] with y 6= : We lso hve tht f : [; b] (; )! R is GH-h-conve on [; b] is equivlent to the fct tht f is k-composite-g h-concve on [; b] with k : (; )! (; ) ; nd g (t) = ln t; t I: By mking use of Corollry 4 we get k (t) = t (4.3) h m y ln y h ( ) ; f (t) dt t h f ( p y) nf y o + ( ) f + y f y + f () + ( ) f (y) [h ( ) + ] f () + [ + ] f (y) for ny [; ] nd for y; [; b] with y 6= : 5. Applictions for HA; HG nd HH-h-Conve Functions ; Let f : [; b] (; )! R be n HA-h-conve function on the intervl [; b] : This is equivlent to the fct tht f is composite-g h-conve on [; b] with the incresing function g (t) = t : Then by pplying Corollry 4 for k (t) = t; we hve the inequlities (5.) h min h ( ) ; y f + y y y h ( ) f + f ( ) + ( + ) y ( ) + y y y y f (t) dt t f y + ( ) f () + f (y) ( ) + y f[h ( ) + ] f () + [ + ] f (y)g for ny [; ] nd for y; [; b] with y 6= : Let f : [; b] (; )! (; ) be n HG-h-conve function on the intervl [; b] : This is equivlent to the fct tht f is k-composite-g h-conve on [; b]

17 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 7 with k : (; )! R; k (t) = ln t nd g (t) = Corollry 4, we hve the inequlities (5.) y h( f ) minf h( ) ; h()g + y f y ( ) + ( + ) y y y ep y f ln f (t) t dt y ( ) + y t ; t [; b] : Then by pplying f [f ()] [f (y)] R h(t)d y h( ) ( ) + y n o R f [h( )+] () f [h()+ ] (y) h(t)dt for ny [; ] nd for y; [; b] with y 6= : Let f : [; b] (; )! (; ) be n HH-h-conve function on the intervl [; b] : This is equivlent to the fct tht f is k-composite-g h-concve on [; b] with k : (; )! (; ) ; k (t) = t nd g (t) = t ; t [; b] : Then by pplying Corollry 4, we hve the inequlities (5.3) h m h ( ) ; f y h ( ) + y y y y f (t) t dt f y ( ) + y y f + y + ( ) f y ( ) + ( + ) y + f () + ( ) f (y) [h ( ) + ] f () + [ + ] f (y) ; for y; [; b] with y 6= : Applictions for p, r-conve nd LogEp conve functions cn lso be provided. However the detils re not presented here. References [] M. W. Alomri, Some properties of h-mn-conveity nd Jensen s type inequlities. Preprints 7, 776 (doi:.944/preprints7.76.v). [] M. W. Alomri nd M. Drus, The Hdmrd s inequlity for s-conve function. Int. J. Mth. Anl. (Ruse) (8), no. 3-6, [3] M. W. Alomri nd M. Drus, Hdmrd-type inequlities for s-conve functions. Int. Mth. Forum 3 (8), no. 37-4, [4] G. A. Anstssiou, Univrite Ostrowski inequlities, revisited. Montsh. Mth., 35 (), no. 3, [5] G. D. Anderson, M. K. Vmnmurthy nd M. Vuorinen, Generlized conveity nd inequlities, J. Mth. Anl. Appl. 335 (7)

18 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v 8 S. S. DRAGOMIR [6] N. S. Brnett, P. Cerone nd S. S. Drgomir, Some new inequlities for Hermite- Hdmrd divergence in informtion theory, in Stochstic Anlysis & Applictions, Vol. 3, Eds. Y. J. Cho, J. K. Kim nd Y. K. Choi, pp. 7-9, Nov Sci. Publishers, 3, ISBN X. Preprint RGMIA Res. Rep. Coll. 5 (), Issue 4, Art. 8. [Online [7] N. S. Brnett, P. Cerone, S. S. Drgomir, M. R. Pinheiro,nd A. Sofo, Ostrowski type inequlities for functions whose modulus of the derivtives re conve nd pplictions. Inequlity Theory nd Applictions, Vol. (Chinju/Msn, ), 9 3, Nov Sci. Publ., Huppuge, NY, 3. Preprint: RGMIA Res. Rep. Coll. 5 (), No., Art. [Online [8] E. F. Beckenbch, Conve functions, Bull. Amer. Mth. Soc. 54(948), [9] M. Bombrdelli nd S. Vrošnec, Properties of h-conve functions relted to the Hermite- Hdmrd-Fejér inequlities. Comput. Mth. Appl. 58 (9), no. 9, [] W. W. Breckner, Stetigkeitsussgen für eine Klsse verllgemeinerter konveer Funktionen in topologischen lineren Räumen. (Germn) Publ. Inst. Mth. (Beogrd) (N.S.) 3(37) (978), 3. [] W. W. Breckner nd G. Orbán, Continuity properties of rtionlly s-conve mppings with vlues in n ordered topologicl liner spce. Universitte "Bbeş-Bolyi, Fcultte de Mtemtic, Cluj-Npoc, 978. viii+9 pp. [] P. Cerone nd S. S. Drgomir, Midpoint-type rules from n inequlities point of view, Ed. G. A. Anstssiou, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, CRC Press, New York [3] P. Cerone nd S. S. Drgomir, New bounds for the three-point rule involving the Riemnn- Stieltjes integrls, in Advnces in Sttistics Combintorics nd Relted Ares, C. Gulti, et l. (Eds.), World Science Publishing,, [4] P. Cerone, S. S. Drgomir nd J. Roumeliotis, Some Ostrowski type inequlities for n-time di erentible mppings nd pplictions, Demonstrtio Mthemtic, 3() (999), [5] G. Cristescu, Hdmrd type inequlities for convolution of h-conve functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Appro. Conveity 8 (), 3. [6] S. S. Drgomir, Ostrowski s inequlity for monotonous mppings nd pplictions, J. KSIAM, 3() (999), [7] S. S. Drgomir, The Ostrowski s integrl inequlity for Lipschitzin mppings nd pplictions, Comp. Mth. Appl., 38 (999), [8] S. S. Drgomir, On the Ostrowski s inequlity for Riemnn-Stieltjes integrl, Koren J. Appl. Mth., 7 (), [9] S. S. Drgomir, On the Ostrowski s inequlity for mppings of bounded vrition nd pplictions, Mth. Ineq. & Appl., 4() (), [] S. S. Drgomir, On the Ostrowski inequlity for Riemnn-Stieltjes integrl R b f (t) du (t) where f is of Hölder type nd u is of bounded vrition nd pplictions, J. KSIAM, 5() (), [] S. S. Drgomir, Ostrowski type inequlities for isotonic liner functionls, J. Inequl. Pure & Appl. Mth., 3(5) (), Art. 68. [] S. S. Drgomir, An inequlity improving the rst Hermite-Hdmrd inequlity for conve functions de ned on liner spces nd pplictions for semi-inner products. J. Inequl. Pure Appl. Mth. 3 (), no., Article 3, 8 pp. [3] S. S. Drgomir, An inequlity improving the rst Hermite-Hdmrd inequlity for conve functions de ned on liner spces nd pplictions for semi-inner products, J. Inequl. Pure Appl. Mth. 3 (), No., Article 3. [4] S. S. Drgomir, An inequlity improving the second Hermite-Hdmrd inequlity for conve functions de ned on liner spces nd pplictions for semi-inner products, J. Inequl. Pure Appl. Mth. 3 (), No.3, Article 35. [5] S. S. Drgomir, An Ostrowski like inequlity for conve functions nd pplictions, Revist Mth. Complutense, 6() (3), [6] S. S. Drgomir, Opertor Inequlities of Ostrowski nd Trpezoidl Type. Springer Briefs in Mthemtics. Springer, New York,. + pp. ISBN:

19 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE 9 [7] S. S. Drgomir, Ostrowski type inequlities for Lebesgue integrl: survey of recent results. Aust. J. Mth. Anl. Appl. 4 (7), no., Art., 83 pp. [Online [8] S. S. Drgomir, Inequlities for generlized nite Hilbert trnsform of conve functions, Preprint RGMIA Res. Rep. Coll. (8), Art. [9] S. S. Drgomir nd C. E. M. Perce, Selected Topics on Hermite- Hdmrd Inequlities nd Applictions, RGMIA Monogrphs,. [Online [3] S. S. Drgomir, New inequlities of Hermite-Hdmrd type for log-conve functions. Khyym J. Mth. 3 (7), no., [3] Drgomir, S. S. Inequlities of Hermite-Hdmrd type for h-conve functions on liner spces. Proyecciones 34 (5), no. 4, [3] S. S. Drgomir, Inequlities of Hermite-Hdmrd type for GA-conve functions, to pper in Annles Mthemtice Silesine, Preprint RGMIA Res. Rep. Coll. 8 (5), Art. 3. [Online [33] S. S. Drgomir, Inequlities of Hermite-Hdmrd type for GG-conve functions, Preprint RGMIA, Reserch Report Collection, 8 (5), Art. 7, 5 pp., [Online [34] S. S. Drgomir, Some integrl inequlities of Hermite-Hdmrd type for GG-conve functions, Mthemtic (Cluj), 59 (8), No -, 7, pp Preprint RGMIA, Reserch Report Collection, 8 (5), Art. 74. [Online [35] S. S. Drgomir, Inequlities of Hermite-Hdmrd type for HA-conve functions, Mroccn J. Pure & Appl. Anlysis, Volume 3 (), 7, Pges 83-. Preprint, RGMIA Res. Rep. Coll. 8 (5), Art. 38. [Online [36] S. S. Drgomir, Inequlities of Hermite-Hdmrd type for HG-conve functions, Probl. Anl. Issues Anl. Vol. 6 (4), No., 7-7. Preprint, RGMIA Res. Rep. Coll. 8 (5), Art. 79. [Online [37] S. S. Drgomir, Inequlities of Hermite-Hdmrd type for HH-conve functions, to pper in Act et Commenttiones Universittis Trtuensis de Mthemtic, Preprint, RGMIA Res. Rep. Coll. 8 (5), Art. 8. [Online [38] S. S. Drgomir, Inequlities of Hermite-Hdmrd type for composite conve functions, Preprint, RGMIA Res. Rep. Coll. (8), Art. [39] S. S. Drgomir, P. Cerone, J. Roumeliotis nd S. Wng, A weighted version of Ostrowski inequlity for mppings of Hölder type nd pplictions in numericl nlysis, Bull. Mth. Soc. Sci. Mth. Romnie, 4(9) (4) (999), [4] S.S. Drgomir nd S. Fitzptrick, The Hdmrd inequlities for s-conve functions in the second sense. Demonstrtio Mth. 3 (999), no. 4, [4] S.S. Drgomir nd S. Fitzptrick,The Jensen inequlity for s-breckner conve functions in liner spces. Demonstrtio Mth. 33 (), no., [4] S. S. Drgomir nd B. Mond, On Hdmrd s inequlity for clss of functions of Godunov nd Levin. Indin J. Mth. 39 (997), no., 9. [43] S. S. Drgomir nd C. E. M. Perce, On Jensen s inequlity for clss of functions of Godunov nd Levin. Period. Mth. Hungr. 33 (996), no., 93. [44] S. S. Drgomir nd C. E. M. Perce, Qusi-conve functions nd Hdmrd s inequlity, Bull. Austrl. Mth. Soc. 57 (998), [45] S. S. Drgomir, J. Peµcrić nd L. Persson, Some inequlities of Hdmrd type. Soochow J. Mth. (995), no. 3, [46] S. S. Drgomir nd Th. M. Rssis (Eds), Ostrowski Type Inequlities nd Applictions in Numericl Integrtion, Kluwer Acdemic Publisher,. [47] S. S. Drgomir nd S. Wng, A new inequlity of Ostrowski s type in L -norm nd pplictions to some specil mens nd to some numericl qudrture rules, Tmkng J. of Mth., 8 (997), [48] S. S. Drgomir nd S. Wng, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd some numericl qudrture rules, Appl. Mth. Lett., (998), 5-9. [49] S. S. Drgomir nd S. Wng, A new inequlity of Ostrowski s type in L p-norm nd pplictions to some specil mens nd to some numericl qudrture rules, Indin J. of Mth., 4(3) (998),

20 Preprints ( NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v S. S. DRAGOMIR [5] A. El Frissi, Simple proof nd re nement of Hermite-Hdmrd inequlity, J. Mth. Ineq. 4 (), No. 3, [5] L. Fejér, Über die Fourierreihen, II, (In Hungrin) Mth. Nturwiss, Anz. Ungr. Akd. Wiss., 4 (96), [5] E. K. Godunov nd V. I. Levin, Inequlities for functions of brod clss tht contins conve, monotone nd some other forms of functions. (Russin) Numericl mthemtics nd mthemticl physics (Russin), 38 4, 66, Moskov. Gos. Ped. Inst., Moscow, 985 [53] H. Hudzik nd L. Mligrnd, Some remrks on s-conve functions. Aequtiones Mth. 48 (994), no.,. [54] E. Kikinty nd S. S. Drgomir, Hermite-Hdmrd s inequlity nd the p-hh-norm on the Crtesin product of two copies of normed spce, Mth. Inequl. Appl. (in press) [55] U. S. Kirmci, M. Klriµcić Bkul, M. E Özdemir nd J. Peµcrić, Hdmrd-type inequlities for s-conve functions. Appl. Mth. Comput. 93 (7), no., [56] M. A. Ltif, On some inequlities for h-conve functions. Int. J. Mth. Anl. (Ruse) 4 (), no. 9-3, [57] D. S. Mitrinović nd I. B. Lcković, Hermite nd conveity, Aequtiones Mth. 8 (985), 9 3. [58] D. S. Mitrinović nd J. E. Peµcrić, Note on clss of functions of Godunov nd Levin. C. R. Mth. Rep. Acd. Sci. Cnd (99), no., [59] P. Montel, Sur les functions convees et les fonctions soushrmoniques, Journl de Mth., 9 (98), 7, 9 6. [6] C. P. Niculescu, Conveity ccording to the geometric men, Mth. Inequl. Appl., 3, (),, [6] C. E. M. Perce, J. Peµcrić nd V. Šimić, Stolrsky mens nd Hdmrd s inequlity. J. Mth. Anl. Appl., 99-9 (998) [6] C. E. M. Perce nd A. M. Rubinov, P-functions, qusi-conve functions, nd Hdmrd-type inequlities. J. Mth. Anl. Appl. 4 (999), no., 9 4. [63] J. E. Peµcrić nd S. S. Drgomir, On n inequlity of Godunov-Levin nd some re nements of Jensen integrl inequlity. Itinernt Seminr on Functionl Equtions, Approimtion nd Conveity (Cluj-Npoc, 989), 63 68, Preprint, 89-6, Univ. "Bbeş-Bolyi, Cluj-Npoc, 989. [64] J. Peµcrić nd S. S. Drgomir, A generliztion of Hdmrd s inequlity for isotonic liner functionls, Rdovi Mt. (Srjevo) 7 (99), 3 7. [65] M. Rdulescu, S. Rdulescu nd P. Alendrescu, On the Godunov-Levin-Schur clss of functions. Mth. Inequl. Appl. (9), no. 4, [66] M.. Sriky, A. Sglm, nd H. Yildirim, On some Hdmrd-type inequlities for h- conve functions. J. Mth. Inequl. (8), no. 3, [67] E. Set, M. E. Özdemir nd M.. Sr ky, New inequlities of Ostrowski s type for s-conve functions in the second sense with pplictions. Fct Univ. Ser. Mth. Inform. 7 (), no., [68] M.. Sriky, E. Set nd M. E. Özdemir, On some new inequlities of Hdmrd type involving h-conve functions. Act Mth. Univ. Comenin. (N.S.) 79 (), no., [69] M. Tunç, Ostrowski-type inequlities vi h-conve functions with pplictions to specil mens. J. Inequl. Appl. 3, 3:36. [7] S. Vrošnec, On h-conveity. J. Mth. Anl. Appl. 36 (7), no., [7] K. S. hng nd J. P. Wn, p-conve functions nd their properties. Pure Appl. Mth. 3(), 3-33 (7). Mthemtics, College of Engineering & Science, Victori University, PO Bo 448, Melbourne City, MC 8, Austrli. E-mil ddress: sever.drgomir@vu.edu.u URL: DST-NRF Centre of Ecellence in the Mthemticl, nd Sttisticl Sciences, School of Computer Science, & Applied Mthemtics, University of the Witwtersrnd,, Privte Bg 3, Johnnesburg 5, South Afric