Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral

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1 DOI 763/s Moroccn J Pure nd Appl AnlMJPAA) Volume ), 6, Pges ISSN: RESEARCH ARTICLE Generlized Hermite-Hdmrd-Fejer type inequlities for GA-conve functions vi Frctionl integrl I mdt I s cn nd Sercn Turhn Astrct In this pper, we give new identity for differentile nd GA-conve functions As result of this identity, we otin some new frctionl integrl inequlities for differentile GA-conve functions Mthemtics Suject Clssifiction Primry 6D5; Secondry 6A5, Third 6D, Fourth 6A5 Key words nd phrses GA-conve, Hermite-Hdmrd-Fejer type inequlities, Frctionl Integrl Introduction The clssicl or the usul conveity is defined s follows: A function f : = 6 I R R, is sid to e conve on I if the inequlity f t t) y) tf ) t) f y) holds for ll, y I nd t, ] A numer of ppers hve een written on inequlities using the clssicl conveity nd one of the most fscinting inequlities in mthemticl nlysis is stted s Received Ferury 6, 6 - Accepted My, 6 c The Authors) 6 This rticle is pulished with open ccess y Sidi Mohmed Ben Adllh University Deprtment of Mthemtics, Fculty of Arts nd Sciences, Giresun University, 8, Giresun, TURKEY e-mil: imdti@yhoocom, imdtiscn@giresunedutr Dereli Voctionl High School, Giresun University, 8, Giresun, TURKEY e-mil: sercnturhn8@gmilcom, sercnturhn@giresunedutr 34

2 follows: ON GA-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRAL 35 ) f f)d f) f), ) where f : I R R e conve mpping nd, I with < Both the inequlities in ) hold in reversed direction if f is concve The inequlities stted in ) re known s Hermite-Hdmrd inequlities For more results on ) which provide new proof, significnt etensions, generliztions, refinements, counterprts, new Hermite-Hdmrd-type inequlities nd numerous pplictions, we refer the interested reder to ]-5],7,,, 4] nd the references therein The usul notion of conve functions hve een generlized in diverse mnners One of them is clled GA-conve functions nd is stted in the definition elow Definition, ]A function f : I R =, ) R is sid to e GA-conve function on I if f λ y λ) λf ) λ) f y) holds for ll, y I nd λ, ], where λ y λ nd λf ) λ) f y) re respectively the weighted geometric men of two positive numers nd y nd the weighted rithmetic men of f) nd fy) The definition of GA-conveity is further generlized s GA-s-conveity in the second sense s follows Definition 4] A function f : I R =, ) R is sid to e GA-s-conve function on I if f λ y λ) λ s f ) λ) s f y) holds for ll, y I nd λ, ] nd for some s, ] For the properties of GA-conve functions nd GA-s-conve function, we refer the reder to 6, 9,, 4, 5, 6] nd the references therein Most recently, numer of findings hve een seen on Hermite-Hdmrd type integrl inequlities for GA-conve nd for GA-s-conve functions Zhng et ll in 5] estlished the following Hermite-Hdmrd type integrl inequlities for GA-conve function Theorem 5] Let f : I R =, ) R e differentile on I, nd, I with < nd f L, ] If f q is GA-conve on, ] for q, then /q f) f) f)d ) A, )] ) /q { L, ) ] f ) q L, ) ] f ) q} /q

3 36 İMDAT İŞCAN AND SERCAN TURHAN Theorem 5] Let f : I R =, ) R e differentile on I, nd, I with < nd f L, ] If f q is GA-conve on, ] for q >, then f) f) f)d ln ln ) 3) L q/q), q/q) ) q/q)] /q A f ) q, f ) q)] /q Theorem 3 5] Let f : I R =, ) R e differentile on I, nd, I with < nd f L, ] If f q is GA-conve on, ] for q > nd q > p >, then /q f) f) f)d ln ln ) 4) p /q L qp)/q), qp)/q) ) ] /q { L p, p ) p ] f ) q p L p, p )] f ) q} /q Theorem 4 6] Suppose tht f : I R =, ) R is GA-s-conve function in the second sense, where s, ) nd let,, ), < If f L, ], then the following inequlities hold the constnt k = s ) s f ln ln f) d f) f), 5) s is the est possile in the second inequlity in ) If f is GA-conve function in Theorem 4, then we get the following inequlities ) f ln ln f) d f) f) 6) For more results on GA-conve function nd GA-s-conve function see eg 6, 9, 4] Definition 3 9] A function f : I R =, ) R is sid to e geometriclly symmetric with respect to if the inequlity ) g = g ) holds for ll, ] Definition 4 8] Let f L, ] The right-hnd side nd left-hnd side Hdmrd frctionl integrls J α f nd J α f of order α > with > re defined y J α f) = Γα) ln ) α dt ft) t t, >

4 ON GA-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRAL 37 J α f) = Γα) ln t ) α ft) dt t, < respectively where Γα) is the Gmm function defined y Γα) = J f) = J f) = f) Lemm 3] For < θ nd < we hve θ θ ) θ e t t α dt nd In 4], D Y Hwng estlished new identity for conve functions In this study, we will prove similr identity nd will otin Hermite-Hdmrd-Fejér inequlity for GA-conve functions vi frctionl integrls sed on this new identity Min Results Throughout in this section, we will use the nottions L t) = t G t, U t) = t G t nd G = G, ) = Lemm Let f : I R =, ) R e differentile function on I o,, I o with < If h :, ], ) is differentile function nd f L, ]), the following inequlity holds: = h) h)] f) h) f) f)h )d 7) ln ln h t G t) h) ] f t G t) t G t dt 4 h t G t) h) ] f t G t) t G t dt Proof We clculte the integrls on the right side of 7), s follows I = h t G t) h) ] d f t G t)) = h t G t) h) ] f t G t) ) ln f t G t) h t G t) t G t dt G

5 38 İMDAT İŞCAN AND SERCAN TURHAN nd I = h t G t) h) ] d f t G t)) Therefore = h t G t) h) ] f t G t) ) ln f t G t) h t G t) t G t dt G I I This completes the proof of the lemm = h) h)] f) h) f) ln ln f t G t) h t G t) t G t dt f t G t) h t G t) t G t dt 8) Theorem Let f : I R =, ) R e differentile function on I o nd, I o with < If h :, ], ) is differentile function nd f is GA-conve on, ], the following inequlity holds f) h) h)] h) f) f)h )d 9) where ln ln 4 ζ, ) f ) ζ, ) f G) ζ 3, ) f ) ], ζ, ) = ζ, ) = t t G t h t G t) h) dt, ) t) t G t h t G t) h) dt )

6 ON GA-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRAL 39 t) t G t h t G t) h) dt nd ζ 3, ) = t t G t h t G t) h) dt Proof We get the following inequlity y tking the solute vlue on oth sides of the equlity in 7): f) h) h)] ln ln 4 h) f) f)h )d h t G t) h) f t G t) t G t dt h t G t) h) f t G t) t G t dt Since f is GA-conve on, ] in ), we hve for ll t, ] tht f) h) h)] ln ln 4 h) f) f)h )d h t G t) h) t f ) t) f G) ] t G t dt h t G t) h) t f ) t) f G) ] t G t dt This completes the proof of the theorem Corollry Suppose tht g :, ], ) is continuous positive mpping nd geometriclly symmetric with respect to ie g ) = g) holds for ll, ] ln ) α ) ] with < ) Choosing h) = t ln t α gt) dt for ll, ] nd t α > in Theorem 5, we otin ) f) f) J α g) J α g)] J α fg) ) J α fg) )] 4) ) 3)

7 4 İMDAT İŞCAN AND SERCAN TURHAN where ln ln )α α Γ α ) g C α) f ) C α) f G) C 3 α) f ) ], C α) = C α) = t) α t) α ] t t G t dt, t) t) α t) α ] t G t t G t] dt nd C 3 α) = t) α t) α ] t t G t dt Specilly, if we use Lemm in 4), for < α, we hve ) f) f) J α g) J α g)] J α fg) ) J α fg) )] 5) where ln ln )α Γ α ) g D α) f ) D α) f G) D 3 α) f ) ] D α) = D α) = t α t G t, t) t α t G t t) t α t G t] dt nd D 3 α) = t α t G t Proof If we tke h) = ) ] α gt) dt for ll, ] in the in- t equlity 9), we hve ln t ) α ln t ) f) f) Γ α) J α g) J α g)] Γ α) J α fg) ) J α fg) )] 6)

8 ON GA-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRAL 4 ln ln 4 t G t ln ) α ) ] ln α g) d ln ) α ) ] ln α g) d t f ) t) f G) ] t G t dt t G t ln ) α ) ] ln α g) d ln ) α ) ] ln α g) d t f ) t) f G) ] t G t dt Since g) is geometriclly symmetric with respect to =, we hve nd = t G t ln ) ] α ln ) α g) d 7) ln ) ] α ln ) α g) d t G t ln ) ] α ln ) α g) d t G t = t G t ln ) ] α ln ) α g) d 8) ln ) ] α ln ) α g) d t G t ln ) ] α ln ) α g) d t G t for ll t, ] By using 7)-8) in 6), we hve ) f) f) J α g) J α g)] J α fg) ) J α fg) )] 9)

9 4 İMDAT İŞCAN AND SERCAN TURHAN ln ln 4Γ α) ln ln 4Γ α) t G t ln ) α ) ] ln α g) d t G t t f ) t) f G) ] t G t dt t G t ln ) α ) ] ln α g) d t G t t f ) t) f G) ] t G t dt ] t G t ln ) α ) ] ln α d g t G t t f ) t) f G) ] t G t dt ] t G t ln ) α ) ] ln α d t G t t f ) t) f G) ] t G t dt In the lst inequlity, we clculte integrls simply s follows: = t G t t G t t G t t G t ln ) ] α ln ) α ln ) α d t G t t G t ln ln )α = t) α t) α ] α α By Lemm, for < α, we hve = t G t t G t t G t t G t ln ) ] α ln ) α d ln ) α d t G t t G t d ) ln ) α d ln ) α d ln ln )α t α α A comintion of 9) nd ), we hve 4) nd 5) Thus the proof is completed Corollry )If we tke α =, we otin the following Hermite-Hdmrd-Fejer type inequlity for GA-conve functions relted to 5):

10 ON GA-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRAL 43 ] f) f) g) d f) g) d ln ln ) g 4 D ) f ) D ) f G) D 3 ) f ) ], where for, >, we hve ) D ) = D ) = t t G t dt = t t) t G t dt { ln ln 4 ln ln t t) t G t dt } 8 8G ln ln ), = D 3 ) = ln ln t t G t dt = { G) ln ln { ln ln } 8 ) ln ln ), 4 ln ln )If we tke g) = in 4), we otin the following inequlity } 8 8G ln ln ) ) f) f) Γα ) ln ln ) J α α f ) J α f )] ) ln ln ) C α α) f ) C α) f G) C 3 α) f ) ], α > 3)If we tke g) = nd α = in 5), we otin the following inequlity ) f) f) ln ln ) ln ln ) 4 f) d D ) f ) D ) f G) D 3 ) f ) ] Theorem Let f : I R =, ) R e differentile function on I o,, I o with < If h :, ], ) is differentile function nd f q is GA-conve on, ] for q, the following inequlity holds f) h) h)] h) f) f)h )d 3) 4)

11 44 İMDAT İŞCAN AND SERCAN TURHAN ln ln 4 ) h t G t q ) h) dt ) h t G t ) h) dt) q t qt G qt) f ) q t) qt G qt) f G) q) ) h t G t q ) h) dt ) h t G t ) h) dt) q t qt G qt) f ) q t) qt G qt) f G) q) Proof Continuing from 3) in proof of Theorem 5, the power men inequlity nd using the fct tht f q is GA-conve on, ], we get the required result This completes the proof of the theorem Corollry 3 Let g :, ], ) e positive continuous mpping nd geometriclly symmetric with respect to ie g ) = g) holds for ll, ] with ln ) α ) ] < ) If h) = t ln t α gt) dt for ll, ], α > in Theorem t 6, we otin ) f) f) J α g) J α g)] J α fg) ) J α fg) )] 5) ln ln )α g α α Γ α ) α ) q C α, q) f ) q C α, q) f G) q C 3 α, q) f ) q] q where for q, α >, C α, q) = C α, q) = t) α t) α ] t qt G qt) dt, t) α t) α ] t) qt G qt) qt G qt)) dt nd C 3 α, q) = t) α t) α ] t qt G qt) dt Proof When we use the equlity ) in 4), we otin ) f) f) J α g) J α g)] J α fg) ) J α fg) )] 6)

12 ON GA-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRAL 45 ln ln )α g α Γ α ) ) t) α t) α q ] dt t) α t) α ] t qt G qt) f ) q t) qt G qt) f G) q) dt ) t) α t) α q ] dt t) α t) α ] t qt G qt) f ) q t) qt G qt) f G) q) dt ln ln )α g α α Γ α ) α ) q ) q t) α t) α ] t qt G qt) f ) q t) qt G qt) f G) q] dt ) q ) q ) t) α t) α ] q t qt G qt) f ) q t) qt G qt) f G) q] dt By using the inequlity r r r ) r for, >, r, we hve ) f) f) J α g) J α g)] J α fg) ) J α fg) )] 7) ln ln )α g α α Γ α ) α ) q t) α t) α ] t qt G qt) f ) q t) α t) α ] t) qt G qt) qt G qt) t) α t) α ] t qt G qt) f ) q Corollry 4 When α = nd g) = ln ln ) f G) q dt re tken in Corollry 3, we otin ) f) f) f) ln ln d ln ln ) C, q) f ) q C, q) f G) q C 3, q) f ) q] q q q 8)

13 46 İMDAT İŞCAN AND SERCAN TURHAN References ] SS Drgomir, Hermite-Hdmrd s type inequlities for conve funtions of selfdjoint opertors in Hilert spces, Liner Alger Appl 436 ), no 5, ] SS Drgomir nd CEM Perce, Selected topics on Hermite-Hdmrd type inequlities nd pplictions, RGMIA Monogrphs, Aville online t hdmrdhtml 3] J Hu, B-Y Xi, nd F Qi, Hermite-Hdmrd type inequlities for geometriclly s-conve functions, CommunKoren MthSoc9 4), No, pp5-63 4] D-Y Hwng, Some inequlities for differentile conve mpping with ppliction to weighted trpezoidl formul nd higher moments of rndom vriles, Applied Mthemtics nd Computtion, 7 ), ] İ İşcn, New estimtes on generliztion of some integrl inequlities for s-conve functions nd their pplictions, Interntionl Journl of Pure nd Applied Mthemtics, 86, No4 3) 6] İ İşcn, Hermite-Hdmrd type inequlities for GA-s-conve functions, Le Mtemtiche, LXIX 4)-Fsc II, pp ] A P Ji, T Y Zhng, F Qi, Integrl Inequlities of Hermite Hdmrd Type α, m)-ga conve Functions, Journl of Function Spce nd Applictions, 3 3), Article ID 83856, 8 pges 8] A A Kils H M Srivstv, J J Trujillo, Theory nd pplictions of frctionl differentil equtionselsevier, Amsterdm 6) 9] MA Ltif, S S Drgomir, E Momonit, Some Fejér type integrl inequlities relted with geometriclly-rithmeticlly-conve functions with pplictions, Sumitted) ] MA Ltif, New Hermite Hdmrd type integrl inequlities for GA-conve functions with pplictions Volume 34, Issue 4 Nov 4) ] C P Niculescu, Conveity ccording to the geometric men, Mth Inequl Appl 3 ) ), Aville online t ] C P Niculescu, Conveity ccording to mens, Mth Inequl Appl 6 4) 3), Aville online t 3] A P Prudnikov, Y A Brychkov, O J Mrichev, Integrl nd series, Elementry Functions, Vol,Nuk, Moscow, 98 4] Y Shung, H-P Yin, nd F Qi, Hermite-Hdmrd type integrl inequlities for geometricrithmeticlly s-conve functions, Anlysis Munich) 33 ) 3), 97-8 Aville online t 5] T-Y Zhng, A-P Ji nd F Qi, Some inequlities of Hermite-Hdmrd type for GA-conve functions with pplictions to mens, Le Mtemtiche, Vol LXVIII 3) Fsc I, pp 9 39 doi: 448/3687 6] X-M Zhng, Y-M Chu, nd X-H Zhng, The Hermite-Hdmrd Type Inequlity of GA- Conve Functions nd Its Appliction, Journl of Inequlities nd Applictions, Volume, Article ID 5756, pges doi:55//5756