New Inequalities of Hermite-Hadamard-like Type for Functions whose Second Derivatives in Absolute Value are Convex

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1 It. Joural of Math. Aalysis, Vol. 8, 1, o. 16, HIKARI Ltd, New Ieualities of Hermite-Hadamard-like Type for Fuctios whose Secod Derivatives i Absolute Value are Covex Jaekeu Park Departmet of Mathematics Haseo Uiversity Seosa, Choogam, , Korea Copyright c 1 Jaekeu Park. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this article, some ew Hadamard-like, which estimate the differece betwee 1 b b a a fxdx ad 1 { f 1ab f a 1b } { 8 b a f 1ab f a 1b} for N with, are established for fuctios whose secod derivative i absolute values are covex. Mathematics Subject Classificatio: 6A51, 6D15 Keywords: Hermite-Hadamard s ieuality, covex fuctio, cocave fuctio, Hölder ieuality, power mea ieuality 1 Itroductio The followig defiitio for covex fuctios is well kow i the mathematical literature: A fuctio f : I R R is said to be covex o I if the ieuality ftx 1 ty tfx1 tfy 1 holds for all x, y I ad t [, 1], ad f is said to be cocave o I if the ieuality 1 holds i reversed directio.

2 778 Jaekeu Park May ieualities have bee established for covex fuctios but the most famous is the Hermite-Hadamard s ieuality, due to its rich geometrical sigificace ad applicatios, which is stated as follow: a b f 1 b a b a fxdx fafb. The double ieualities hold i reversed directio if f is cocave. The ieuality of Hermite-Hadamard type has bee cosidered the most useful ieuality i mathematical aalysis. Some of the classical ieualities for meas ca be derived from for particular choices of the fuctio f. I [5], Dragomir ad Agarwal established the followig result which give estimate betwee the middle ad the rightmost terms i : Theorem 1.1. Let f : I R R be a differetiable fuctio o the iterior I of a iterval I ad f L[a, b], where a, b I with a<b. If f p is covex o [a, b] for some fixed p>1, the the followig ieuality holds: fafb 1 b a b a fxdx p 1 1 p b a [ f p a p 1 f p b p 1 ] p 1 p. 3 I [13], Pecaric ad Pečarić established the followig result which give estimate betwee the middle ad the rightmost terms i : Theorem 1.. Let f : I R R be a differetiable fuctio o the iterior I of a iterval I ad f L[a, b], where a, b I with a<b. If f is covex o [a, b] for some fixed 1, the the followig ieuality holds: fafb 1 b a b a fxdx b a [ f a f b ] 1. For more results o Hermite-Hadamard-type ieuality providig ew proofs, oteworthy extesios, geeralizatio ad umerous applicatios, see [3]-[17] ad refereces therei. I recet paper [15], Tseg et al. established the followig result which gives a refiemet of : f a b 1 { f 3a b a 3b } f 1 b a b a fxdx 1 [ f a b fafb ] fafb, 5 where f :[a, b] R is a covex fuctio I [11], Latif ad Dragomir established the followig results which give estimate betwee the middle ad { 1 f 3ab f a3b } for fuctios whose derivatives i absolute value are covex:

3 New ieualities of Hermite-Hadamard-like type 779 Theorem 1.3. Let f : I R R be a differetiable fuctio o the iterior I of a iterval I ad f L[a, b], where a, b I with a<b. If f is covex o [a, b], the the followig ieuality holds: 1 { b a 96 f 3a b a 3b } f 1 b a [ f a { f 3a b b a fxdx f a 3b } f a b ] f b 6 I [], Ciurdariu established the followig ieualities for fuctios whose twice derivative i absolute values are covex; Theorem 1.. Let f : I R R be a twice differetiable fuctio o the iterior I of a iterval I ad f L[a, b], where a, b I with a<b. If f is covex o [a, b] for some fixed >1with 1 1 =1, the the followig p ieuality holds: 1 { f 3a b a 3b } f 1 fxdx b a a b a 1 1 p 1 1 [ f Cp, l 1 { 3a b f a } 1 18 p 1 { f Cp, l a b f 3a b 1 } { f Cp, l 3 a 3b f a b 1 } { f Cp, l a 3b f b 1 } ], where Cp, l = 1 p l 1 p p l 1 p. I this article, the mai aim is to establish some ew Hadamard-like, which estimate the differece betwee 1 b fxdx ad { 1 b a a f 1ab f a 1b } { 8 b a f 1ab f a 1b} for N with, are established for fuctios whose twice derivative i absolute values are covex ad s-covex i the secod sese. b Mai results I order to prove our mai theorems, we eed the followig lemma, which is a geeralizatio of Lemma 1 give by Ciurdariu[]:

4 78 Jaekeu Park Lemma 1. Let f : I R be a twice differetiable fuctio o the iterior I of a iterval I i R where a, b I with a<b.iff L 1 [a, b] for N with, the the followig idetity holds: Ifa, b; 1 b fxdx 1 { 1a b a 1b} f f b a a { b a f 1a b f a 1b} 8 b [{ a = t f 1a b t 1 ta 3 t 1 f a 1b } tb 1 t 3 { t f a 1b t 1 t a b t 1 f t a b 1a b }] 1 t. Proof. By itegratio by parts ad by makig use of the substitutio x = t 1ab 1 ta, we have a = t f 1a b t 1 ta 3 { b a f 1a b b a b a f 1a b 1ab a } fxdx. 7 Aalogously, we also have the followig eualities: b t 1 f a 1b tb 1 t 3 { b a f = a 1b b a b a f a 1b b a 1b } fxdx. 8

5 New ieualities of Hermite-Hadamard-like type 781 c t f a 1b t 1 t a b 3 { b a f = a 1b b a b a f a 1b a 1b d t 1 f t a b 1a b 1 t 3 { b a f = 1a b b a b a f 1a b ab ab 1ab } fxdx. 9 } fxdx. 1 Addig 7-1, we get the desired idetity, which completes the proof. We will use this lemma for obtaiig several followig results similar to Theorem 1- from [15]. Theorem.1. Let f : I R be a twice differetiable fuctio o the iterior I of a iterval I i R ad f L[a, b], where a, b I with a<b. If f is a covex fuctio o [a, b], the the followig ieuality holds: Ifa, b; b [{ a f a f b 3 f 3 1a b f a 1b } 3 { f a b f 3 a 1b f 1a b }]. Proof. From Lemma 1, we get Ifa, b; b [{ a 3 t f 1a b t 1 ta

6 78 Jaekeu Park 3 { 1 t f tb 1 t t f a 1b t a 1b } 1 t a b 1 t f t a b 1a b }] 1 t. 11 Usig the covexity of f o [a, b], we observe that the followig ieuality holds: a t f 1a b t 1 ta 1 f 1a b 1 f a, 1 1 b 1 t f a 1b tb 1 t 1 f b 1 f a 1b, 13 1 c t f a 1b t 1 t a b 1 f a 1b 1 f a b, 1 1 d 1 t f t a b 1a b 1 t 1 f a b 1 f 1a b By substitutig 1-15 i 11, we easily get the desired result, which is a geeralizatio of Theorem 1.3. Corollary.1. Suppose that all the coditios of Theorem.1 are satisfied. The we have Ifa, b; b a 6 1 { f a f b }. 8 Moreover, if f M for all x [a, b], the we have also the followig ieuality Ifa, b; b a 6 1 M.

7 New ieualities of Hermite-Hadamard-like type 783 Corollary.. I Theorem.1, if we choose =, the we have b [ a f Ifa, b; a f b f a b 1536 { f 6 3a b f a 3b }] b { } a f a f b. 19 Theorem.. Let f : I R be a twice differetiable fuctio o the iterior I of a iterval I i R ad f L[a, b], where a, b I with a<b.if f is a covex fuctio o [a, b] for some fixed >1, the the followig ieuality holds: Ifa, b; b a 1 p 3 p 1 [{ f Cp, l 1 1a b f a 1 f Cp, l a 1b f b 1 } 3 { f Cp, l 3 a 1b f a b 1 f Cp, l 1a b f a b 1 }], where Cp, l i i =1,, 3, are defied as i Theorem 1.. Proof. From Lemma 1 ad usig the well-kow Hölder itegral ieuality[1], we get Ifa, b; b a 1 1 p 3 p 1 [{ Cp, l 1 f 1a b t Cp, l f tb 1 t 3 { Cp, l 3 f a 1b t 1 t a b Cp, l f t a b 1a b 1 t 1 1 ta a 1b 1 } 1 1 }]. 16

8 78 Jaekeu Park Usig the covexity of f o [a, b], we observe that the followig ieuality holds: a f 1a b t 1 ta c d 1 { f 1a b b f tb 1 t 1 f t 1 f a }, 17 a 1b { f b f a 1b }, 18 a 1b 1 t a b { f a 1b f a b }, 19 f t a b 1a b 1 t 1 { f a b f 1a b }. By substitutig 17- i 16, we easily get the desired result, which is a geeralizatio of Theorem 1.. Corollary.3. Suppose that all the coditios of Theorem. are satisfied. The we have Ifa, b; b a 1 p 3 p 1 [{ 1 Cp, l 1 f a 1 f b 1 1 Cp, l f a 1 f b 1 } 3 { Cp, l 3 f a 3 f b 1 3 Cp, l f a f b 1 }].

9 New ieualities of Hermite-Hadamard-like type 785 Corollary.. I Theorem., if we choose =, the we obtai the ieuality: Ifa, b; b a 1 [ p f Cp, l 1 3a b f a 1 56 p 1 Cp, l f a 3b f b 1 f Cp, l 3 a 3b f a b 1 f Cp, l 3a b f a b 1 ]. Theorem.3. Let f : I R be a twice differetiable fuctio o the iterior I of a iterval I i R ad f L[a, b], where a, b I with a<b.if f is a covex fuctio o [a, b] for some fixed 1, the the followig ieuality holds: Ifa, b; b a [{ f 1a b 1 f 3 a 1 f a 1b 1 f 3 b 1 } 3 { f a 1b 1 f a b 1 3 f 1a b 1 f a b 1 }]. 3 Proof. From Lemma 1 ad usig the well-kow power mea ieuality, we get Ifa, b; b a [{ t f 1a b 1 t 1 ta t f a 1b 1 } tb 1 t 3 { t f a 1b t 1 t a b 1 1 t f t a b 1a b 1 }] 1 t. 1

10 786 Jaekeu Park Usig the covexity of f o [a, b], we observe that the followig ieuality holds: a t f 1a b t 1 ta b c 1 f 1a b 1 f a, 1 1 t f a 1b tb 1 t 1 f b 1 f a 1b, 3 1 t f a 1b t 1 t a b d 1 f a 1b 1 f a b, 1 1 t f t a b 1a b 1 t 1 f a b 1 f 1a b. 5 1 By substitutig -5 i 1, we easily get the desired result. Corollary.5. Suppose that all the coditios of Theorem.3 are satisfied. The we have Ifa, b; b a [{ 3 f a 1 f b f a 3 f b 1 } 3 3 { 6 f a 7 6 f b f a 6 f b 1 }]. 6 6

11 New ieualities of Hermite-Hadamard-like type 787 Corollary.6. I Theorem.3, if we choose =, the we obtai the ieuality: Ifa, b; b a [ f 3a b 1 f a 1 3 f a 3b 1 f b 1 3 f a 3b 1 f a b 3 1 f 3a b 1 f a b 3 1 ]. Theorem.. Let f : I R be a twice differetiable fuctio o the iterior I of a iterval I i R ad f L[a, b], where a, b I with a<b.if f is a cocave fuctio o [a, b] for some fixed >1, the the followig ieuality holds: Ifa, b; b a 1 1 [{ Cp, l 3 1 f 1a b 3 1 Cp, l f a 1b 1 } 3 { f Cp, l 3 a 3 b Cp, l f 3 a b 1 }]. Proof. From Lemma 1 ad usig the power mea ieuality ad the Hölder itegral ieuality[1], we get Ifa, b; b a [{ Cp, l 1 f 1a b 1 t 1 ta Cp, l f a 1b 1 } tb 1 t 3 { Cp, l 3 f a 1b t 1 t a b 1 Cp, l f t a b 1a b 1 }] 1 t. 6

12 788 Jaekeu Park Sice f is cocave o [a, b], by substitutig x = t 1ab ad usig the ieuality b 1 fxdx f b a a we have the followig ieualities: a f 1a b t a b, 1 ta 1 ta c d b f 1a b, 7 f a 1b tb 1 t f a 1b, 8 f a 1b t 1 t a b f a 3 b, 9 f t a b 1a b 1 t f 3 a b. 3 By substitutig 7-3 i 6, we easily get the desired result. Theorem.5. Let f : I R be a twice differetiable fuctio o the iterior I of a iterval I i R ad f L[a, b], where a, b I with a<b.if f is a cocave fuctio o [a, b] for some fixed 1, the the followig ieuality holds: Ifa, b; b a [{ f 3a 3b f 3a 3b } { f 6a 7 6b 8 f 7 6a 6b 8 }].

13 New ieualities of Hermite-Hadamard-like type 789 Proof. First, by the cocavity of f o [a, b] ad the power-mea ieuality, we have which implies that f tx 1 ty t fx 1 t fy t fx 1 t fy, f tx 1 ty t fx 1 t fy for all t [, 1] ad x, y [a, b]. Accordigly, usig Lemma 1 ad usig the Jese s itegral ieuality, we get Ifa, b; b [{ a f 6 3 t t 1ab 1 ta t f 1 t tb 1 t a 1b } 1 t 3 { f t t a 1b 1 t ab 1 t Note that f 1 t t ab 1 t 1ab }] 1 t b a f t t 1ab 1 ta t 31 = f 3a 3b, 3 f 1 t tb 1 t a 1b 1 t c = f 3a 3b, 33 f t t a 1b 1 t ab t

14 79 Jaekeu Park d = f 6a 7 6b, 3 8 f 1 t t ab 1 t 1ab 1 t = f 7 6a 6b By substitutig 3-35 i 31, we easily get the desired result. Corollary.7. I Theorem.5, if we choose =, the we obtai the ieuality: Ifa, b; b a [ f 13a 3b f 3a 13b f 5a 11b f 11a 5b ] Refereces [1] Z. Chagjia, M. Becze, O Hölder s ieuality ad its applicatios, Creative Math. If., 181 9, [] L. Ciurdariu, O some Hermite-Hadamard type ieualities for fuctios whose power of absolute value of derivatives are α, m-covex, It. J. of Math. Aal., 68 1, [3] S. S. Dragomir, C. E. M. Pearce, Selected topic o Hermite-Hadamard ieualities ad applicatios, Melboure ad Adelaide December,. [] S. S. Dragomir, S. Fitzpatrick, The Hadamard s ieuality for s-covex fuctios i the secod sese, Demostratio Math., , [5] S. S. Dragomir, R. P. Agarwal, Two ieualities for differetiable mappigs ad applicatios to special meas of real umbers ad to trapezoidal formula, Appl. Math. Lett., , [6] H. Hudzik, L. Maligrada, Some remarks o s-covex fuctios, Aeuatioes Math., 8 199, [7] H. Kavurmaci, M. Avci, M. E. Özdemir, New ieualities of Hermite-Hadamard type for covex fuctios with applicatios, arxiv: v1[math.CA].

15 New ieualities of Hermite-Hadamard-like type 791 [8] U. S. Kirmaci, Ieualities for differetiable mappigs ad applicatios to special meas of real umbers ad to midpoit formula, Appl. Math. Comput., 171, [9] U. S. Kirmaci, K. Klaričić Bakula, M. E. Özdemir, J. Pečarić, Hadamarype ieualities for s-covex fuctios, Appl. Math. Comput., , [1] U. S. Kirmaci, M. E. Özdemir, O some ieualities for differetiable mappigs ad applicatios to special meas of real umbers ad to midpoit formula, Appl. Math. Comput., 153, [11] M. A. Latif, S. S. Dragomir, New ieualities of Hermite-Hadamard type for fuctios whose derivatives i absolute value are covex with applicatios, Acta Uiv. Matthiae Belii, Series Math., 13 13, [1] B. G. Pachpatte, O some ieualities for covex fuctios, RGMIA Res. Rep. Coll., 6E 3. [13] C. E. M. Pecaric, J. Pečarić, Ieualities for differetiable mappigs with applicatios to special meas ad uadrature formula, Appl. Math. Lett., 13, [1] E. Set, New ieualities of Ostrowski type for mappigs whose derivatives are s-covex i the secod via fractioal itegrals, Comput. Math. Appl., 1 Art ID:531976, 7 pages. [15] K. L. Tseg, R. S. Hwag, S. S. Dragomir, Fejér-type ieualities I, J. Ieuel. ad Appl., 13 1, 13 pp. [16] M. Tuç, O some ew ieualities for covex fuctios, Turk. J. Math., 35 11, 1-7. [17] M. Tuç, New itegral ieualities for s-covex fuctios, RGMIA Res. Rep. Coll., 13 1, Received: February 9, 1