Parametrized inequality of Hermite Hadamard type for functions whose third derivative absolute values are quasi convex

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1 Wu et l. SpringerPlus (5) 4:83 DOI.8/s z RESEARCH Prmetrized inequlity of Hermite Hdmrd type for functions whose third derivtive bsolute vlues re qusi convex Shn He Wu, Bnyt Sroysng, Jin Shn Xie nd Yu Ming Chu 3* Open Access *Correspondence: 3 School of Mthemtics nd Computtion Science, Hunn City University, Yiyng 43, Hunn, People s Republic of Chin Full list of uthor informtion is vilble t the end of the rticle Abstrct In this pper we present some inequlities of Hermite-Hdmrd type for functions whose third derivtive bsolute vlues re qusi-convex. Moreover, n ppliction to specil mens of rel numbers is lso considered. Keywords: Hermite-Hdmrd type inequlity, Prmeter, Qusi-convex, Specil mens Mthemtics Subject Clssifiction: D, A5, E Bckground A rel-vlued function f defined on n intervl I R is sid to be convex on I, if f (λx + ( λ)y) λf (x) + ( λ)f (y) for ll x, y I nd λ [, ]. If f is convex on I, then we hve the Hermite-Hdmrd inequlity (see Mitrinović et l. 993) ( ) + b f f (x)dx f () + f (b) for ll, b I. A function f : I R R is sid to be qusi-convex on I, if f (λx + ( λ)y) mx { f (x), f (y) } for ll x, y I nd λ [, ]. Clerly, ny convex function is qusi-convex function. Furthermore, there exist qusi-convex functions which re not convex. In 7, Ion (7) presented n inequlity of Hermite-Hdmrd type for functions whose derivtives in bsolute vlues re qusi-convex functions, s follows: () 5 Wu et l. This rticle is distributed under the terms of the Cretive Commons Attribution 4. Interntionl License ( cretivecommons.org/licenses/by/4./), which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthor(s) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Wu et l. SpringerPlus (5) 4:83 Pge of 9 Theorem. Let f : I R R be differentible mpping on I o,, b I with < b. If f is qusi-convex on [, b], then the following inequlity holds: f () + f (b) f (x)dx mx { f (), f (b) }. 4 () In, Alomri et l. () estblished n nlogous version of inequlity (), which is sserted by Theorem. below: Theorem. Let f : I R R be twice differentible mpping on I o,, b I with < b nd f is integrble on [, b]. If f is qusi-convex on [, b], then the following inequlity holds: f () + f (b) f (x)dx () mx { f (), f (b) }. (3) Recently, Guo et l. (5) investigted Hermite-Hdmrd type inequlities for geometriclly qusi-convex functions. Xi nd Qi (4, 5) nd Xi et l. (, 4) showed some new Hermite-Hdmrd type inequlities for s-convex functions. For more results relting to refinements, counterprts, generliztions of Hdmrd type inequlities, we refer interested reders to Alomri et l. (b), Chen (5), Niculescu nd Persson (), Pečrić et l. (99), Sroysng (4), Sroysng (3) nd Wu (9). The min purpose of this pper is to present prmetrized inequlity of Hermite- Hdmrd type for functions whose third derivtive bsolute vlues re qusi-convex. As pplictions, some new inequlities for specil mens of rel numbers re estblished. Lemms In order to prove our min results, we need the following lemms. Lemm. Let ǫ R nd let f: I R R be three times differentible on I nd, b I with < b. Assume tht f is integrble on [, b]. Then (4 ǫ)f () + ( + ǫ)f (b) f (x)dx ( (b) ( ǫ)f () ) ()3 λ( λ)(λ ǫ)f (λ + ( λ)b)dλ.

3 Wu et l. SpringerPlus (5) 4:83 Pge 3 of 9 Proof Integrting by prts, we hve λ( λ)(λ ǫ)f (λ + ( λ)b)dλ λ( λ)(λ ǫ)df (λ + ( λ)b) b λ [ λ( λ)(λ ǫ)f (λ + ( λ)b) ] λ b λ f (λ + ( λ)b)d(λ( λ)(λ ǫ)) b λ f (λ + ( λ)b)d(λ( λ)(λ ǫ)) b () λ λ λ ( λ + ( + ǫ)λ ǫ ) f (λ + ( λ)b)dλ ( ) λ ( + ǫ)λ + ǫ df (λ + ( λ)b) [( () λ ( + ǫ)λ + ǫ () λ ( () (b) ( ǫ)f () ) ) ] λ f (λ + ( λ)b) λ ( f (λ + ( λ)b)d λ ( + ǫ)λ + ǫ () (λ ( + ǫ))f (λ + ( λ)b)dλ λ ( () (b) ( ǫ)f () ) λ + () 3 (λ ǫ)df (λ + ( λ)b) λ ( () (b) ( ǫ)f () ) + [(λ ǫ)f (λ + ( λ)b)]λ () 3 λ () 3 λ f (λ + ( λ)b)d(λ ǫ) ( () (b) ( ǫ)f () ) ( ) + (4 ǫ)f () + ( + ǫ)f (b) () 3 () 3 λ f (λ + ( λ)b)dλ. )

4 Wu et l. SpringerPlus (5) 4:83 Pge 4 of 9 Chnging vrible x λ + ( λ)b, it follows tht Thus, λ( λ)(λ ǫ)f (λ + ( λ)b)dλ ( () (b) ( ǫ)f () ) ( ) + (4 ǫ)f () + ( + ǫ)f (b) () 3 () 3 () 4 xb x f (x)dx. λ( λ)(λ ǫ)f (λ + ( λ)b)dλ (4 ǫ)f () + ( + ǫ)f (b) The proof of Lemm. is completed. f (x)dx ( (b) ( ǫ)f () ). Lemm. Let ǫ be rel number. Then Proof ǫ 4ǫ λ( λ) λ ǫ dλ 3 ǫ 4 48 ǫ We distinguish three cses + ǫ if ǫ if <ǫ< if ǫ. Cse If ǫ, then Cse If <ǫ<, then λ( λ) λ ǫ dλ Cse 3 If ǫ, then λ( λ) λ ǫ dλ λ( λ) λ ǫ dλ ǫ/ 4ǫ3 ǫ 4 48 λ( λ)(ǫ λ)dλ ǫ. λ( λ)(ǫ λ)dλ + λ( λ)(λ ǫ)dλ ǫ/ + ǫ. λ( λ)(λ ǫ)dλ ǫ. This completes the proof of Lemm..

5 Wu et l. SpringerPlus (5) 4:83 Pge 5 of 9 Min results Our min results re stted in the following theorems. Theorem 3. Let q nd ǫ R, nd let f : I R R be three times differentible on I nd, b I with < b. Assume tht f is integrble on [, b], nd f q is qusiconvex on [, b]. Then (4 ǫ)f () + ( + ǫ)f (b) f (x)dx ( (b) ( ǫ)f () ) () 3 ( ) ǫ (mx { f () q, f (b) q}) /q () 3 ( 4ǫ 3 ǫ 4 + ǫ ) (mx { f () q, f (b) q}) /q 48 () 3 ( ) ǫ (mx { f () q, f (b) q}) /q if ǫ if <ǫ< if ǫ. Proof Using Lemm. nd Hölder s inequlity gives (4 ǫ)f () + ( + ǫ)f (b) ()3 ()3 By the qusi-convexity of f q, we obtin (4 ǫ)f () + ( + ǫ)f (b) f (x)dx ( (b) ( ǫ)f () ) ( ) /q ()3 λ( λ) λ ǫ dλ ( λ( λ) λ ǫ mx { f () q, f (b) ) /q q} dλ λ( λ) λ ǫ f (λ + ( λ)b) dλ (λ( λ) λ ǫ ) /q ( λ( λ) λ ǫ f (λ + ( λ)b) q) /q dλ ( ) /q ()3 λ( λ) λ ǫ dλ ( λ( λ) λ ǫ f (λ + ( λ)b) /q dλ) q. ()3 f (x)dx ( (b) ( ǫ)f () ) ( (mx { λ( λ) λ ǫ dλ) f () q, f (b) q}) /q.

6 Wu et l. SpringerPlus (5) 4:83 Pge of 9 Utilizing Lemm. leds to the desired inequlity in Theorem 3.. Remrk 3. It is worth noticing tht if we use substitution b, b nd ǫ ǫ in Theorem 3., we hve the following further generliztion of Theorem 3.. Theorem 3.3 Let q nd ǫ R, nd let f : I R R be three times differentible on I,, b I with b. Assume tht f is integrble on [, b], nd f q is qusi-convex on the closed intervl formed by the points nd b. Then (4 ǫ)f () + ( + ǫ)f (b) f (x)dx ( (b) ( ǫ)f () ) 3 ( ) ǫ (mx { f () q, f (b) q}) /q 3 ( 4ǫ 3 ǫ 4 + ǫ ) (mx { f () q, f (b) q}) /q 48 3 ( ) ǫ (mx { f () q, f (b) q}) /q if ǫ if <ǫ< if ǫ. As direct consequence, choosing ǫ in Theorem 3.3, we get the following inequlity: Corollry 3.4 Let q nd ǫ R, nd let f : I R R be three times differentible on I,, b I with b. Assume tht f is integrble on [, b], nd f q is qusi-convex on the closed intervl formed by the points nd b. Then f () + f (b) f (x)dx ( f (b) f () ) 3 ( { mx f () q, f (b) q}) /q. 9 In ddition, if we utilize Theorem 3. with substitution of ǫ,.5, 3,, 3, 5, respectively, then we obtin the following results: Corollry 3.5 Let q nd ǫ R, nd let f : I R R be three times differentible on I,, b I with b. Assume tht f is integrble on [, b], nd f q is qusi-convex on the closed intervl formed by the points nd b. Then f () + f (b) f (x)dx + f () 3 3 ( { mx f () q, f (b) q}) /q, 7 7f () + 5f (b) f (x)dx ( f (b) 3f () ) ( { mx f () q, f (b) q}) /q, 9

7 Wu et l. SpringerPlus (5) 4:83 Pge 7 of 9 f () + 5f (b) f (x)dx ( 3f (b) + f () ) 3 ( { mx f () q, f (b) q}) /q, 3 f () f (x)dx + ( f (b) + f () ) 3 ( { mx f () q, f (b) q}) /q, 4 7f () f (b) f (x)dx + ( 3f (b) + 5f () ) 3 ( { mx f () q, f (b) q}) /q, 8 3f () f (b) Applictions to specil mens We now consider the pplictions of our results to the specil mens of rel numbers. The weighted rithmetic men of rel numbers {, b } with weight {w, w b } is defined by where, b, w, w b R with w + w b. In prticulrly f (x)dx + ( 5f (b) + 7f () ) 3 ( { mx f () q, f (b) q}) /q. A(, b; w, w b ) w + w b b w + w b, A(, b;, ) A(, b) + b, which is clled the rithmetic mens. The generlized logrithmic men of rel numbers {, b } is defined by [ b n+ n+ ] n L n (, b), (n + )() where, b, R, n Z with n,, b.

8 Wu et l. SpringerPlus (5) 4:83 Pge 8 of 9 Proposition 4. Let, b, ǫ R, b, ǫ nd n N, n 3. Then, we hve A(n, b n ; 4 ǫ,+ ǫ) Proof We consider the function f (x) x n, x R, n 3. It is esy to verify tht the function f (x) n(n )(n )x n3 is qusi-convex on (, + ) (see Alomri et l. b). The ssertion follows from Theorem 3.3 with q. Remrk 4. In similr wy s the proof of the Proposition 4., one cn esily deduce from Corollry 3.4 the following inequlity. Proposition 4.3 A(n, b n ) ()(ǫ )n A( n, b n ; ǫ, ǫ) L n n (, b) 3 ( { (ǫ )n(n )(n ) mx n3, b n3}) if ǫ 7 3 ( 8 8ǫ + 4ǫ 3 ǫ 4) ( { n(n )(n ) mx n3, b n3}) if <ǫ< 57 3 ( { ( ǫ)n(n )(n ) mx n3, b n3}) if ǫ. 7 Let, b R, b nd n N, n 3. Then, we hve () n(n )L n n (, b) Ln n (, b) ( 3 n(n )(n ) 9 mx { n3, b n3}). Conclusions This pper provides some new results relted to the Hermite-Hdmrd type inequlities. Firstly, we present prmetrized inequlity of Hermite-Hdmrd type for functions whose third derivtive bsolute vlues re qusi-convex, the min results re given in Theorems 3. nd 3.3. As specil cses, by ssigning specil vlue to the prmeter, one cn obtin severl new nd previously known results for Hermite-Hdmrd type inequlity (Corollries 3.4 nd 3.5). Secondly, s pplictions of the obtined results, we estblish two new inequlities involving specil mens of rel numbers by using the prmetrized Hermite-Hdmrd type inequlity (see Propositions 4. nd 4.3). Authors contributions SW finished the proof nd the writing work. BS nd JX gve SW some dvice on the proof nd writing. YC gve SW lots of help in revising the pper. All uthors red nd pproved the finl mnuscript. Author detils Deprtment of Mthemtics, Longyn University, Longyn 34, Fujin, People s Republic of Chin. Deprtment of Mthemtics nd Sttistics, Fculty of Science nd Technology, Thmmst University, Pthumthni, Thilnd. 3 School of Mthemtics nd Computtion Science, Hunn City University, Yiyng 43, Hunn, People s Republic of Chin. Acknowledgements The uthors would like to express their herty thnks to nonymous referees for their vluble comments on this rticle. This reserch ws supported by the Nturl Science Foundtion of Fujin province under Grnt 5J5 nd the Outstnding Young Incubtion Progrmme nd Key Project of Fujin Province Eduction Deprtment under Grnt JA499. Competing interests The uthors declre tht they hve no competing interests.

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