FRACTIONAL HERMITE-HADAMARD INEQUALITIES FOR SOME CLASSES OF DIFFERENTIABLE PREINVEX FUNCTIONS

Transcription

1 U.P.B. Sci. Bull., Serie A, Vol. 78, I. 3, 6 ISSN 3-77 FRACTIONAL HERMITE-HADAMARD INEQUALITIES FOR SOME CLASSES OF DIFFERENTIABLE PREINVEX FUNCTIONS Muhammad Alam NOOR, Khalida Inaya NOOR, Marcela V. MIHAI 3, Muhammad Uzair AWAN 4 The main ojecive of hi aricle i o ealih ome new ineualiie of Hermie-Hadamard ype via Riemann-Liouville fracional inegral. We ealih a new fracional inegral ideniy for differeniale funcion, hen uing hi ideniy a auiliary reul we derive ome fracional Hermie-Hadamard ype ineualiie for differeniale -preinve funcion and for differeniale -Godunova-Levin preinve funcion. Keyword: Hermie-Hadamard ineualiie, -preinve funcion, -Godunova-Levin preinve funcion, Riemann-Liouville fracional inegral MSC : 6D5, 6A5. Inroducion The relaionhip eween heory of conve funcion and heory of ineualiie ha inpired many reearcher o inveigae hee heorie. A very inereing reul in hi regard i due o Hermie and Hadamard independenly ha i Hermie-Hadamard ineualiy. Thi remarkale reul of Hermie and Hadamard can e viewed a neceary and ufficien condiion for a funcion o e conve. Thi famou reul read a follow: Le f : I R R e a conve funcion wih a< and a, I. Then a f( a) f( ) f f( ) d. a (.) a For ome ueful deail on Hermie-Hadamard ype ineualiie, ee [,4,5,7,9-3,5-]. Recenly many reearcher have eended he claical concep of conve Deparmen of Mahemaic, COMSATS Iniue of Informaion Technology, Park Road, Ilamaad, Pakian Deparmen of Mahemaic, COMSATS Iniue of Informaion Technology, Park Road, Ilamaad, Pakian 3 Deparmen of Mahemaic, Univeriy of Craiova, Sree A. I. Cuza 3, Craiova, RO-585, Romania 4 Deparmen of Mahemaic, COMSATS Iniue of Informaion Technology, Park Road, Ilamaad, Pakian, Correponding Auhor, awan.uzair@gmail.com

2 64 Muhammad Alam Noor, Khalida Inaya Noor, Marcela V. Mihai, Muhammad Uzair Awan funcion. A a reul many new and inereing generalizaion of claical conve funcion can e found in he lieraure, ee [3-6,5,7,8,9,]. An imporan eenion of conve funcion wa he inroducion of preinve funcion []. For ome ueful and inereing inveigaion on preinve funcion, ee [,, 9,,, 4, 5, 8, 9, ]. Noor [9] and Noor [8] eended he cla of preinve funcion and inroduced he concep of -preinve funcion and -Godunova-Levin preinve funcion repecively. In hi paper, we derive a new fracional inegral ideniy for differniale funcion. Uing hi we oain our main reul ha are fracional Hermie-Hadamard ype ineualiie for differeniale -preinve funcion and differeniale -Godunova-Levin funcion. Some pecial cae are alo deduced. Thi i he main moivaion of hi paper. Preliminary Reul n Le K e a nonempy cloed e in R. Le f : K R e a coninuou n funcion and le η (.,.) : K K R e a coninuou ifuncion. Fir of all, we recall ome known reul and concep. Definiion. []. A e K i aid o e inve e wih repec o η (.,.), if u η (, v u) K, u, v K, [,]. (.) The inve e K i alo called η -conneced e. Remark. []. We would like o menion ha Definiion. of an inve e ha a clear geomeric inerpreaion. Thi definiion eenially ay ha here i a pah aring from a poin u which i conained in K. We do no reuire ha he poin v hould e one of he end poin of he pah. Thi oervaion play an imporan role in our analyi. Noe ha, if we demand ha v hould e an end poin of he pah for every pair of poin uv, K, hen η(, vu)= v u, and coneuenly inveiy reduce o conveiy. Thu, i i rue ha every conve e i alo an inve e wih repec o η(, vu)= v u, u he convere i no necearily rue, ee [4, ] and he reference herein. For he ake of impliciy, we alway aume ha K =[ u, u η( v, u)], unle oherwie pecified. Definiion.3 [] A funcion f i aid o e preinve wih repec o arirary ifuncion η (.,.), if f( u η ( v, u)) ( ) f( u) f ( v), u, v K, [,]. (.) The funcion f i aid o e preconcave if and only if f i preinve. For η(, vu)= v u in (.) he preinve funcion ecome conve

3 Fracional Hermie-Hadamard ineualiie for ome clae of differeniale preinve funcion 65 funcion in he claical ene. Noe ha every conve funcion i a preinve funcion. However i i known [] ha preinve funcion may no e conve funcion. Definiion.4 [9]. A funcion f : K R i aid o e -preinve of econd kind wih repec o η (.,.), if f( u η ( v, u)) ( ) f( u) f( v), u, v K, [,], (,). Noe ha for = he definiion of -preinve funcion reduce o he definiion of preinve funcion. And for η( a, )= a, hen we have he definiion of -Breckner conve funcion. Definiion.5 [8]. A funcion f : K R i aid o e -Godunova-Levin preinve of econd kind wih repec o η (.,.), if f( u) f( v) f( u η ( v, u)), u, v K, (,), [,]. (.3) ( ) I i oviou ha for =, -Godunova-Levin preinve funcion of econd kind reduce o he definiion of P -preinve funcion [8] and for = i reduce o he definiion of Godunova-Levin preinve funcion [8]. When η(, vu)= v u hen we have definiion of -Godunova-Levin funcion of econd kind [4, 5]. Definiion.6 [8]. Le f L[ a, ]. Then Riemann-Liouville inegral J f a and J f of order > wih a are defined y J f( )= ( ) f( )d, > a, a Γ( ) a ( )= ( ) ( )d, <, and J f f Γ( ) where Γ ( )= e d, i he well known Gamma funcion. We now give he definiion of hypergeomeric erie which will e ued in he oaining ome inegral. Definiion.7 [8]. For he real or comple numer ac,,, oher han,,,, he hypergeomeric erie i defined y m a z a( a ) ( ) z ( a) m( ) m z F[ a,, c; z]= =. c! c( c )! ( c) m! m= Here ( φ ) m i he Pochhammer ymol, which i defined y m

4 66 Muhammad Alam Noor, Khalida Inaya Noor, Marcela V. Mihai, Muhammad Uzair Awan m =, ( φ) m = φφ ( ) ( φ m), m>, which ha he inegral form: c a F[ a, ; c; z]= ( ) ( z) d B( c, ) where z <, c > > and y B( y, ) = ( ) d, i Euler funcion Bea wih Γ( ) Γ( y) B( y, ) =. Γ ( y) Now we prove he following auiliary reul which play a key role in proving our main reul. Lemma.8. Le f : K R e differeniale funcion uch ha f Laa [, η( a, )]. Then for >, we have Φ ( ; a,, )( f) η ( a, ) = { f ( a η(, a))d f ( a η(, a))d } η( a, ) η (, ) { f ( η(, ))d f ( η(, ))d }, η( a, ) where Φ ( ; a,, )( f) η (, a) f( a η(, a)) η (, ) f( η(, )) η (, a) f( a) η (, ) f( ) = η(, a) η(, a) Γ ( ) [ J f( a η(, a)) J f( a η(, a)) η(, a) [ a η (, a)] a J ( (, )) ( (, ))]. f η J f η [ η (, )] Proof. I uffice o how ha f ( a η(, a))d

5 Fracional Hermie-Hadamard ineualiie for ome clae of differeniale preinve funcion 67 Γ ( ) = ( η(, )) ( η(, ))d f a a f a a η( a, ) η( a, ) Γ( ) a η (, a) Γ ( ) = ( η(, )) ( η(, )) ( )d η( a, ) η ( a, ) Γ( ) a η (, a ) f a a u a a f u u ( ) = ( (, )) ( (, )). (, ) f a a Γ (, ) J η f a η a a [ a (, a)] a η η η (.4) Similarly Γ ( ) f ( a η(, a))d = f( a) J f( a η(, a)), η( a, ) η ( a, ) a (.5) Γ ( ) η η(, ) η (, ) f ( (, ))d = f( ) J f( η(, )), (.6) and Γ ( ) η(, ) η (, ) [ η (, )] f ( η(, ))d = f( η(, )) J f( η(, )), (.7) Afer uiale rearrangemen he proof i complee. Remark.9. We would like o remark ha for η( a, )= a Lemma.8 reduce o Lemma [3]. 3 Main Reul In hi ecion, we derive our main reul. Theorem 3.. Le f : K R e differeniale funcion uch ha f Laa [, η( a, )]. If f i i -preinve funcion of econd kind, hen, for >, we have Φ ( ; a,, )( f ) ϑ ϑ [ η (, a) () a { η (, a) η (, )} () η (, ) () ], η( a, ) where

6 68 Muhammad Alam Noor, Khalida Inaya Noor, Marcela V. Mihai, Muhammad Uzair Awan Γ ( ) Γ ( ) ϑ = ( ) d = (3.) Γ ( ) ( ) ϑ = ( ) d = F[,,, ]. (3.) Proof. Uing Lemma.8, aking modulu and he fac ha f i -preinve funcion of econd kind, we have: Φ ( ; a,, )( f) η ( a, ) { f ( a η(, a)) d f ( a η(, a)) d } η( a, ) η (, ) { f ( η(, )) d f ( η(, )) d } η( a, ) η ( a, ) { [ f ( a) f ( ) ]d η( a, ) [ f ( a) ( ) ]d } η (, ) { [ f ( ) f ( ) ]d η( a, ) [ f ( ) ( ) ]d } ϑ ϑ a a a η( a, ) = [ η (, ) ( ) { η (, ) η (, )} ( ) η (, ) ( ) ]. Thi complee he proof. Theorem 3.. Le f : K R e differeniale funcion uch ha f Laa [, η( a, )]. If f i i -Godunova-Levin preinve funcion of econd kind, hen, for >, we have Φ ( ; a,, )( f) ϕ ϕ [ η (, a) ( a) { η (, a) η (, )} ( ) η (, ) ( ) ], η( a, ) where

7 Fracional Hermie-Hadamard ineualiie for ome clae of differeniale preinve funcion 69 Γ( ) Γ ( ) ϕ = ( ) d = (3.3) Γ( ) ϕ = ( ) d =( ) F[,,, ]. (3.4) Theorem 3.3. Le f : K R e differeniale funcion uch ha f Laa [, η( a, )] and >. If f i -preinve funcion of econd kind where < < and =, >, hen p Φ ( ; a,, )( f) where p η a μ a μ ( ) { (, ){( ( ) ( ) ) p η( a, ) a ( μ ( ) μ ( ) ) }}, ( μ ( ) μ ( ) ) } η (, ){( μ ( ) μ ( ) ) μ = ( ) d = μ = ( ) d =. Proof. Uing Lemma.8, Hölder ineualiy and he fac ha -preinve funcion of econd kind Φ ( ; a,, )( f ) f i η ( a, ) { f ( a η(, a)) d f ( a η(, a)) d } η( a, ) η (, ) { f ( η(, )) d f ( η(, )) d } η( a, ) p p η ( a, ) (( )d)[( ( a η(, a)) d) η( a, )

8 7 Muhammad Alam Noor, Khalida Inaya Noor, Marcela V. Mihai, Muhammad Uzair Awan ( f ( a η(, a)) d ) ] p p η (, ) (( )d)[( ( η(, )) d) η( a, ) ( f ( η(, )) d ) ] p η ( a, ) ( ) {( [( ) ( a) ( ) ( ) ]d ) p η( a, ) ( [( ) ( a) ( ) ( ) ]d ) } p η (, ) ( ) {( [( ) ( ) ( ) ( ) ]d ) p η( a, ) ([( ) () ( ) () ]d)} p η a μ a μ = ( ) { (, ){( ( ) ( ) ) p η( a, ) a ( μ ( ) μ ( ) ) }}. ( μ ( ) μ ( ) ) } η (, ){( μ ( ) μ ( ) ) Thi complee he proof. Theorem 3.4. Le f : K R e differeniale funcion uch ha f Laa [, η( a, )] and >. If f i -Godunova-Levin preinve funcion of econd kind where < < and =, >, hen p Φ ( ; a,, )( f) p η a λ a λ ( ) { (, ){( ( ) ( ) ) p η( a, )

9 Fracional Hermie-Hadamard ineualiie for ome clae of differeniale preinve funcion 7 a ( λ ( ) λ ( ) ) } η (, ){( λ ( ) λ ( ) ) where λ ( λ ( ) ( ) ) }}, λ = ( ) d = ( ) λ = ( ) d =. Theorem 3.5. Le f : K R e differeniale funcion uch ha f Laa [, η( a, )] and >. If f i -preinve funcion of econd kind, where < < and >, hen Φ ( ; a,, )( f ) η a ϑ a ϑ ( ) { (, ) {( ( ) ( ) ) η( a, ) a ( ϑ ( ) ϑ ( ) ) }}, ( ϑ ( ) ϑ ( ) ) } η(, ) {( ϑ ( ) ϑ ( ) ) where ϑ and ϑ are given y (3.) and (3.). Proof. Uing Lemma.8, Power mean ineualiy and he fac ha i -preinve funcion of econd kind Φ ( ; a,, )( f ) f η ( a, ) { f ( a η(, a)) d f ( a η(, a)) d } η( a, ) η (, ) { f ( η(, )) d f ( η(, )) d } η( a, ) ( a, ) η ( d ) [( ( a η(, a)) d ) η( a, )

10 7 Muhammad Alam Noor, Khalida Inaya Noor, Marcela V. Mihai, Muhammad Uzair Awan ( ( a η(, a)) d ) ] (, ) η ( d ) [( ( η(, )) d ) η( a, ) ( ( η(, )) d ) ] η a ϑ a ϑ ( ) { (, ) {( ( ) ( ) ) η( a, ) a ( ϑ ( ) ϑ ( ) ) }}. ( ϑ ( ) ϑ ( ) ) } η(, ) {( ϑ ( ) ϑ ( ) ) Thi complee he proof. Theorem 3.6. Le f : K R e differeniale funcion uch ha f Laa [, η( a, )] and >. If f i -Godunova-Levin preinve funcion of econd kind, where < < and >, hen Φ ( ; a,, )( f ) η a ϕ a ϕ ( ) { (, ) {( ( ) ( ) ) η( a, ) a ( ϕ ( ) ϕ ( ) ) }}, ( ϕ ( ) ϕ ( ) ) } η(, ) {( ϕ ( ) ϕ ( ) ) where ϕ and ϕ are given y (3.3) and (3.4). Acknowledgemen The auhor would like o hank edior and anonymou referee. The auhor alo would like o hank Dr. S. M. Junaid Zaidi, Recor, COMSATS Iniue of Informaion Technology, Pakian, for providing ecellen reearch and academic environmen. Thi reearch i uppored y HEC NRPU projec No: -966/R&D/-553.

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12 74 Muhammad Alam Noor, Khalida Inaya Noor, Marcela V. Mihai, Muhammad Uzair Awan [] T. Weir, B. Mond, Preinve funcion in muliojecive opimizaion, J. Mah. Anal. Appl. 36(988), [] X. M. Yang, X. Q. Yang, K. L. Teo, Generalized inveiy and generalized invarian monooniciy, J. Opim. Theory. Appl., 7(3),