Hermite-Hadamard and Simpson-like Type Inequalities for Differentiable p-quasi-convex Functions

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1 Filomt 3: htts://doi.org/.98/fil79945i Pulished y Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: htt:// Hermite-Hdmrd nd Simson-like Tye Ineulities for Differentile -usi-conve Functions İm İşcn Sercn Turhn Selhttin Mden c Giresun University Dertment of Mthemtics Fculty of Arts nd Sciences 8 Giresun Turkey. Giresun University Dereli Voctionl School 8 Giresun Turkey. c Ordu University Dertment of Mthemtics Fculty of Arts nd Sciences 5 Ordu Turkey. Astrct. In this er we give new concet which is generliztion of the concets usi-conveity nd hrmoniclly usi-conveity nd estlish new identity. A conseuence of the identity is tht we otin some new generl ineulities contining ll of the Hermite-Hdmrd nd Simson-like tye for functions whose derivtives in solute vlue t certin ower re -usi-conve. Some lictions to secil mens of rel numers re lso given.. Introduction Let f : I R R e conve function defined on the intervl I of rel numers nd I with <. The following ineulity f + d f + f holds. This doule ineulity is known in the literture s Hermite-Hdmrd integrl ineulity for conve functions. Note tht some of the clssicl ineulities for mens cn e derived from for rorite rticulr selections of the ming f. Both ineulities hold in the reversed direction if f is concve. Following ineulity is well known in the literture s Simson ineulity: Theorem.. Let f : [ ] R e four times continuously differentile ming on nd f 4 = su f 4 <. Then the following ineulity holds: [ ] f + f + + f 3 d 88 f 4 4. Mthemtics Suject Clssifiction. Primry 6A5; Secondry 6D5 Keywords. -usi-conve functions Hermite-Hdmrd tye ineulity Simson ineulity Received: 5 Setemer 6; Acceted: 8 July 7 Communicted y Mrko D. Petković Emil ddresses: imi@yhoo.com im.iscn@giresun.edu.tr İm İşcn sercnturhn8@gmil.com sercn.turhn@giresun.edu.tr Sercn Turhn mden55@mynet.com Selhttin Mden

2 İ. İşcn et l. / Filomt 3: The notion of usi-conve functions generlizes the notion of conve functions. More recisely function f : [ ] R is sid usi-conve on [ ] if f α + αy su { f y } for ny y [ ] nd α [ ]. Clerly ny conve function is usi-conve function. Furthermore there eist usi-conve functions which re not conve see []. For some results which generlize imrove nd etend the ineulities relted to usi-conve functions we refer the reder to see [ 4 6 5] nd lenty of references therein. In [5] the uthor gve the definition of hrmoniclly conve function s follow nd estlished Hermite- Hdmrd s ineulity for hrmoniclly conve functions. Definition.. Let I R\ {} e rel intervl. A function f : I R is sid to e hrmoniclly conve if y f t f y + t t + ty for ll y I nd t [ ]. If the ineulity in is reversed then f is sid to e hrmoniclly concve. In [5] Zhng et l. defined the hrmoniclly usi-conve function nd sulied severl roerties of this kind of functions. Definition.3. A function f : I [ is sid to e hrmoniclly conve if y f su { f y } t + ty for ll y I nd t [ ]. We would like to oint out tht ny hrmoniclly conve function on I is hrmoniclly usi-conve function ut not conversely. For emle the function ]; = [ 4]. is hrmoniclly usi-conve on 4] ut it is not hrmoniclly conve on 4]. In [9] Zhng nd Wn gve definition of -conve function s follow: Definition.4. Let I e -conve set. A function f : I R is sid to e -conve function or elongs to the clss PCI if f [ α + αy ] / α + α f y for ll y I nd α [ ]. Remrk.5 [9]. An intervl I is sid to e -conve set if [ α + αy ] / I for ll y I nd α [ ] where = k + or = n/m n = r + m = t + nd k r t N. Remrk.6 [7]. If I e rel intervl nd R\ {} then [ α + αy ] / I for ll y I nd α [ ]. According to Remrk.6 we cn give different version of the definition of -conve function s follow:

3 İ. İşcn et l. / Filomt 3: Definition.7. Let I e rel intervl nd R\ {}. A function f : I R is sid to e -conve function if f [ α + αy ] / α + α f y for ll y I nd α [ ]. According to Definition.7 It cn e esily seen tht for = nd = -conveity reduces to ordinry conveity nd hrmoniclly conveity of functions defined on I resectively. In [6 Theorem 5] if we tke I R\ {} nd ht = t then we hve the following Theorem. Theorem.8. Let f : I R e -conve function R\ {} nd I with <. If f L[ ] then we hve [ ] + / f f + f d. 3 For some results relted to -conve functions nd its generliztions we refer the reder to see [ ].. Min Results Definition.. Let I e rel intervl nd R\ {}. A function f : I R is sid to e -usi-conve if f [ t + ty ] / m { f y } 4 for ll y I nd t [ ]. If the ineulity in 4 is reversed then f is sid to e -usi-concve. It cn e esily seen tht for r = nd r = -usi conveity reduces to ordinry usi conveity nd hrmoniclly usi conveity of functions defined on I resectively. Morever every -conve function is -usi-conve function. Emle.. Let f : R = R\ {} nd : R = c c R then f nd re -usi-conve functions. Proosition.3. Let I e rel intervl R\ {} nd f : I R is function then ;. if nd f is usi-conve nd nondecresing function then f is -usi-conve.. if nd f is -usi-conve nd nondecresing function then f is usi-conve. 3. if nd f is -usi-concve nd nondecresing function then f is usi-concve. 4. if nd f is usi-concve nd nondecresing function then f is -usi-concve. 5. if nd f is usi-conve nd nonincresing function then f is -usi-conve. 6. if nd f is -usi-conve nd nonincresing function then f is usi-conve. 7. if nd f is -usi-concve nd nonincresing function then f is usi-concve. 8. if nd f is usi-concve nd nonincresing function then f is -usi-concve. Proof. Since = [ is conve function on nd = ] is concve function on the roof is ovious from the following ower men ineulities [ t + ty ] / t + ty nd [ t + ty ] / t + ty.

4 The following roosition is ovious. İ. İşcn et l. / Filomt 3: Proosition.4. If f : [ ] R nd if we consider the function : [ ] R defined y t = f t / then f is -usi-conve on [ ] if nd only if is usi-conve on [ ]. In order to rove our min results we need the following lemm: Lemm.5. Let f : I R e differentile function on I nd I with <. If f L[ ] nd R\ {} then for λ [ ] we hve the eulity λ f f + f M d = / λ t f λ t M + f M where M = M = [t + t ] / nd M / = M. Proof. It suffices to note tht I = / λ t f M = t λ f / M / f M = λ f / M f f M. Setting = t + t nd d = which gives / I = λ f M f M d. Similrly we cn show tht Thus I = / λ t f M = λ f + λ f M M d. I + I = λ f f + f M d which is reuired.

5 İ. İşcn et l. / Filomt 3: Theorem.6. Let f : I R e differentile function on I I with < nd f L[ ]. If f is -usi-conve on [ ] for nd R\ {} then we hve the following ineulity for λ [ ] λ f M f + f d 5 Cλ; ; + Cλ; ; m { f f } / where for = ϑm u ϑ Cλ; ; u ϑ = ln ϑ u ϑ + ϑ + M λ u ϑ ϑ + M u ϑ M λ u ϑ u nd for R\ { } Cλ; ; u ϑ = M λ + ϑ u [ ϑ + + M + u ϑ M + u ϑ λ + ϑ + M λ u ϑ ϑ + M u ϑ M λ u ϑ] u ϑ > nd M = [t + t ] / nd M / = M. Proof. From Lemm.5 nd using the ower men integrl ineulity we hve λ f M f + f d / λ t f M λ t + f M + / / / λ t λ t / / / / λ t f M λ t M. Hence y -usi conveity of f on [ ] we hve λ f M f + f d / λ t / λ t m { f f } λ t + λ t m { f f }

6 It is esily check tht nd / İ. İşcn et l. / Filomt 3: Cλ; ; + Cλ; ; m { f f } / λ t = Cλ; ; / λ t = Cλ; ;. This concludes the roof. In Theorem.6 if we tke = then we otin the following result for usi-conve functions. Corollry.7. Under the ssumtions Theorem.6 with = we hve + f + f λ f d Cλ; ; + Cλ; ; m { f f } /. In Theorem.6 if we tke = then we otin the following result for hrmoniclly usi-conve functions. Corollry.8. Under the ssumtions Theorem.6 with = we hve λ f + f + f d Cλ; ; + Cλ; ; m { f f } /. Corollry.9. Under the ssumtions Theorem.6 with = we hve λ f M f + f d Cλ; ; + Cλ; ; m { f f }. 6 Corollry.. Under the ssumtions Theorem.6 with λ = we hve f M d C; ; + C; ; m { f f } /

7 İ. İşcn et l. / Filomt 3: where for = [ ] M u ϑ C; ; u ϑ = ln + ϑ M ϑ u u ϑ ϑ ϑ nd for R\ { } C; ; u ϑ = + ϑ u [ M + u ϑ ϑ + + ϑ M u ϑ ϑ ]. Corollry.. Under the ssumtions Theorem.6 with λ = we hve f + f d where for = [ ϑ C; ; u ϑ = ln ϑ u M u ϑ nd for R\ { } C; ; u ϑ = C; ; + C; ; m { f f } / + ] ϑ + M λ u ϑ ϑ M u ϑ + ϑ u [ ϑ + M + u ϑ + ϑ + M λ u ϑ ϑ M u ϑ ]. Corollry.. Under the ssumtions Theorem.6 with λ = /3 we hve [ f + f + f M ] 3 d C/3; ; + C/3; ; m { f f } / wherewhere for = ϑm u ϑ C/3; ; u ϑ = ln ϑ u ϑ u nd for R\ { } C/3; ; u ϑ = M 6 + ϑ + M [ ϑ + + M + + ϑ u u ϑ M + u ϑ 6 ϑ + ϑ + M u ϑ + M u ϑ M u ϑ]. 3 6 ϑ u ϑ + M u ϑ M 3 6 u ϑ 3. Some Alictions for Secil Mens Let us recll the following secil mens of two nonnegtive numer with > :. The rithmetic men A = A := +.

8 İ. İşcn et l. / Filomt 3: The geometric men G = G :=. 3. The hrmonic men H = H := The Logrithmic men L = L := ln ln. 5. The -Logrithmic men L = L := 6. The Identric men I = I = e 7. The ower men + + M = M = +. R\ { }. + / R\ {}. These mens re often used in numericl roimtion nd in other res. However the following simle reltionshis re known in the literture: H G L I A. It is lso known tht L is monotoniclly incresing over R denoting L = I nd L = L. Proosition 3.. Let < < R\ { / } nd λ [ ]. Then we hve the following ineulity λ M + M + + L L + Cλ; ; + Cλ; ; m { } where Cλ; ; is defined s in Theorem.6. Proof. The ssertion follows from the ineulity 6 in Corollry.9 for f : R = + /+. Proosition 3.. Let < < nd λ [ ]. Then we hve the following ineulity λ ln A ln G ln I where Cλ; ; is defined s in Theorem.6. Cλ; ; + Cλ; ; Proof. The ssertion follows from the ineulity 6 in Corollry.9 for = nd f : R = ln. 4. Conflict of Interests The uthor declres tht there is no conflict of interests regrding the uliction of this er.

9 İ. İşcn et l. / Filomt 3: References [] D. A. Ion Some estimtes on the Hermite-Hdmrd ineulity through usi-conve functions Annls of University of Criov Mth. Com. Sci. Ser [] İ. İşcn Generliztion of different tye integrl ineulities vi frctionl integrls for functions whose second derivtives solute vlues re usi-conve Konurl journl of Mthemtics [3] İ. İşcn New generl integrl ineulities for usi-geometriclly conve functions vi frctionl integrls Journl of Ineulities nd Alictions ges. [4] İ. İşcn On generliztion of some integrl ineulities for usi-conve functions nd their lictions Interntionl Journl of Engineering nd Alied sciences EAAS [5] İ. İşcn Hermite-Hdmrd tye ineulities for hrmoniclly conve functions. Hcet. J. Mth. Stt [6] İ. İşcn nd S. Numn Ostrowski tye ineulities for hrmoniclly usi-conve functions Electronic Journl of Mthemticl Anlysis nd Alictions July [7] İ. İşcn Ostrowski tye ineulities for -conve functions Reserchgte doi:.34/rg Aville online t htts:// [8] İ. İşcn Hermite-Hdmrd tye ineulities for -conve functions Reserchgte doi:.34/rg Aville online t htts:// [9] K. S. Zhng J.P. Wn -conve functions nd their roerties. Pure Al. Mth [] M. W. Alomri M. Drus U. S. Kirmci Refinements of Hdmrd-tye ineulities for usi-conve functions with lictions to trezoidl formul nd to secil mens Comuters nd Mthemtics with Alictions [] M. Alomri S. Hussin Two ineulities of Simson tye for usi-conve functions nd lictions Alied Mthemtics E-Notes -7. [] M. A. Noor K. I. Noor M. V. Mihi nd M. U. Awn Hermite-Hdmrd ineulities for differentile -conve functions using hyergeometric functions Pulictions De L Institut Mthémtiue Nouvelle Série tome [3] M. V. Mihi M. A. Noor K. I. Noor nd M. U. Awn New estimtes for trezoidl like ineulities vi differentile h-conve functions Reserchgte doi:.34/rg Aville online t htts:// [4] M. A. Noor K. I. Noor nd S. Iftikhr Nonconve Functions nd Integrl Ineulities Punj University Journl of Mthemtics [5] T.-Y. Zhng A.-P. Ji nd F. Qi Integrl ineulities of Hermite-Hdmrd tye for hrmoniclly usi-conve functions. Proc. Jngjeon Mth. Soc [6] Z. B. Fng nd R. Shi On the h-conve function nd some integrl ineulities J. Ineul. Al ges.