INEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI A, M. DARUS A, AND S.S. DRAGOMIR B Astrct. In this er, some ineulities of Hermite-Hdmrd tye for functions whose detivtives solute vlues re usi-convex, re given. Some error estimtes for the midoint formul re lso otined.. Introduction Let f I R! R e convex function de ned on the intervl I of rel numers nd I, with <. The following ineulity, known s the Hermite Hdmrd ineulity for convex functions, holds (.) f f () + f () In recent yers mny uthors hve estlished severl ineulities connected to Hermite-Hdmrd s ineulity. For recent results, re nements, counterrts, generliztions nd new Hermite-Hdmrd-tye ineulities see [] [5] nd [7] []. In [], Drgomir nd Agrwl otined ineulities for di erentile convex mings which re connected with Hermite-Hdmrd s ineulity nd they used the following lemm to rove it. Lemm. Let f I R! R e di erentile ming on I where I with <. If f L[ ], then the following eulity holds (.) f () + f () = Z The min ineulity in [] is ointed out s follows ( t) f (t + ( t) ) dt Theorem. Let f I R! R e di erentile ming on I, where I with <. If jf j is convex on [ ], then the following ineulity holds Z f () + f () (.3) [jf ()j + jf ()j] In [] Perce nd Peµcrić using the sme Lemm roved the following theorem. Key words nd hrses. Convex function, Hermite-Hdmrd ineulity, Qusi-convex functions. The nncil suort received from Universiti Kengsn Mlysi, Fculty of Science nd Technology under the grnt no. (UKM GUP TMK 7 7) is grtefully cknowledged.
M. ALOMARI, M. DARUS, AND S.S. DRAGOMIR Theorem. Let f I R! R e di erentile ming on I, where I with <. If jf j is convex on [ ], for some, then the following ineulity holds (.) nd (.5) f () + f () f jf ()j + jf ()j jf ()j + jf ()j If jfj is concve on [ ] for some, then Z f () + f () (.6) f nd (.7) f f In [7] some ineulities of Hermite-Hdmrd tye for di erentile convex mings were roved using the following lemm Lemm. Let f I R! R e di erentile ming on I where I with <. If f L[ ], then the following eulity holds Z Z (.) f = ( ) K (t) f (t + ( t) ) dt where, K (t) = ( t t t t One more generl result relted to (.7) ws estlished in []. The min result in [7] is s follows Theorem 3. Let f I R! R e di erentile ming on I, where I with <. If jf j is convex on [ ], then the following ineulity holds Z (.9) f [jf ()j + jf ()j] Now, we recll tht the notion of usi-convex functions generlizes the notion of convex functions. More recisely, function f [ ]! R is sid usi-convex on [ ] if f (x + ( ) y) mx ff (x) f (y)g for ny x y [ ] nd [ ] Clerly, ny convex function is usi-convex function. Furthermore, there exist usi-convex functions which re not convex (see [6]). Recently, D.A. Ion [6] estlished two ineulities for functions whose rst derivtives in solute vlue re usi-convex. Nmely, he otined the following results
INEQUALITIES OF HERMITE-HADAMARD S TYPE 3 Theorem. Let f I R! R e di erentile ming on I, I with <. If jf j is usi-convex on [ ], then the following ineulity holds Z f () + f () (.) mx fjf ()j jf ()jg Theorem 5. Let f I R! R e di erentile ming on I, I with <. If jf j is usi-convex on [ ], then the following ineulity holds Z f () + f () (.) ( ) ( + ) n mx jf ()j jf ()j o The min urose of this er is to estlish ineulities relted to the left hnd side of Hermite-Hdmrd s tye for functions whose derivtives in solute vlue re usi-convex. The otined results cn e used to give estimtes for the roximtion error of the integrl R y the use of the midoint formul.. Hermite-Hdmrd Tye Ineulities Let us strt with n imrovement nd simli ction of the constnts in Theorem 5 nd consolidte this result with Theorem. Theorem 6. Let f I R! R e di erentile ming on I, I with <. If jf j is usi-convex on [ ],, then the following ineulity holds Z f () + f () (.) su jf ()j jf ()j Proof. From Lemm, using the well-known ower men ineulity, we hve Z f () + f () Z = ( t) f (t + ( t) ) dt Z j tj jf (t + ( t) )j dt Z j Z j mx jf ()j jf ()j Z tj dt j tj jf (t + ( t) )j dt tj dt mx Z jf ()j jf ()j j tj dt Corollry. Let f e s in Theorem 6. Additionlly, if
M. ALOMARI, M. DARUS, AND S.S. DRAGOMIR () jf j is incresing, then we hve f () + f () (.) () jf j is decresing, then we hve f () + f () (.3) jf ()j jf ()j Remrk. For = this reduces to Theorem. For = =( ) ( > ) we hve n imrovement of the constnts in Theorem 5, since > + if > nd ccordingly < ( + ) Next, our min result(s) resent new ineulities of midoint tye for usiconvex functions. Theorem 7. Let f I R! R e di erentile ming on I, I with <. If jf j is usi-convex on [ ], then the following ineulity holds Z (.) f mx f jf ()j + mx f jf ()j Proof. From Lemm, we hve Z f " Z ( ) t jf (t + ( t) )j dt + " Z ( ) t mx f Z + ( t) mx Z jf ()j dt j tj jf (t + ( t) )j dt # jf ()j f dt mx f jf ()j + mx f # jf ()j In the following, we deduce nd imrove some ineulities of Hermite-Hdmrd tye. Corollry. Let f e s in Theorem 7. Additionlly, if () jf j is incresing, then we hve Z (.5) f jf ()j + f
INEQUALITIES OF HERMITE-HADAMARD S TYPE 5 () jf j is decresing, then we hve (.6) f jf ()j + f (3) f + =, then we hve (.7) f [jf ()j + jf ()j] () f () = f () =, then we hve (.) f f Proof. It follows directly y Theorem 7. Similr result(s) re emodied in the following theorem. Theorem. Let f I R! R e di erentile ming on I, I with <. If jf j =( ) is usi-convex on [ ], >, then the following ineulity holds (.9) Z ( ( ) mx ( + ) f + mx ( f )! =( ) jf ()j =( ) =( ) f jf ()j =( ) )! 3 5
6 M. ALOMARI, M. DARUS, AND S.S. DRAGOMIR Proof. From Lemm, using well known Hölder integrl ineulity, we hve Z f " Z Z # ( ) t jf (t + ( t) )j dt + j tj jf (t + ( t) )j dt ( ) ( ) Z t dt + ( ) Z t dt + ( )! Z Z ( t) dt! Z Z ( t) dt " ( ) = mx f ( + ) + mx f jf (t + (! Z mx f! Z t) )j dt! jf (t + ( t) )j dt! jf ()j dt mx jf ()j jf ()j #! jf ()j f! dt where + =, which comletes the roof. Corollry 3. Let f e s in Theorem. Additionlly, if () jf j =( ) is incresing, then we hve Z ( ) (.) f jf ()j + ( + ) f () jf j =( ) is decresing, then we hve Z ( ) (.) f jf ()j + ( + ) f (3) f + =, then we hve Z ( ) (.) f [jf ()j + jf ()j] ( + ) () f () = f () =, then we hve Z ( ) (.3) f ( + ) f An imrovement of the constnts in Theorem nd consolidte this result with Theorem 7 is s follows
INEQUALITIES OF HERMITE-HADAMARD S TYPE 7 Theorem 9. Let f I R! R e di erentile ming on I, I with <. If jf j is usi-convex on [ ],, then the following ineulity holds Z (.) f " mx f jf ()j # + mx f jf ()j Proof. From Lemm, using the well-known ower men ineulity, we hve Z f (.5) ( ) ( ) + ( ) Z Z t jf (t + ( t) )j dt + Z tdt! Z ( t) dt Since jf j is usi-convex we hve nd Z Z t jf (t + ( Z t jf (t + (! Z ( t) jf (t + ( t) )j dt t) )j dt t) )j dt mx f! ( t) jf (t + ( t) )j dt jf ()j j tj jf (t + ( t) )j dt jf mx ()j f Therefore, we hve Z f " mx f + mx jf ()j f! # jf ()j Remrk. For = this reduces to Theorem 7. For = =( ) ( > ) we hve n imrovement of the constnts in Theorem, since > + if > nd ccordingly < ( + )
M. ALOMARI, M. DARUS, AND S.S. DRAGOMIR Imrovements of the ineulities (.5), (.6), (.7) nd (.) re given in the following result Corollry. Let f e s in Theorem 9. Additionlly, if () jf j is incresing, then (.5) holds. () jf j is decresing, then (.6) holds. (3) f + =, then (.7) holds. () f () = f () =, then (.) holds. Proof. Follows directly from Theorem 9. 3. Alictions to the Midoint Formul Let d e division of the intervl [ ], i.e., d = x < x < < x n < x n =, nd consider the midoint formul nx xi + x i+ (3.) M (f d) = (x i+ x i ) f i= It is well known tht if the ming f [ ]! R, is di erentile such tht f (x) exists on ( ) nd K = su x() jf (x)j <, then (3.) I = = M (f d) + E (f d) where the roximtion error E (f d) of the integrl I y the midoint formul M (f d) stis es (3.3) je (f d)j K nx (x i+ x i ) 3 i= It is cler tht if the ming f is not twice di erentile or the second derivtive is not ounded on ( ), then (3.3) cnnot e lied. In the following, we roose some new estimtes for the reminder term E(f d) in terms of the rst derivtive which re etter thn the estimtions of [7, ] nd []. Proosition. Let f I R! R e di erentile ming on I, I with <. If jf j is usi-convex on [ ], then in (3.), for every division d of [ ], the following holds (3.) je (f d)j nx (x i+ x i ) mx f xi + x i+ i= + mx f xi + x i+ jf (x i+)j jf (x i )j
INEQUALITIES OF HERMITE-HADAMARD S TYPE 9 Proof. Alying Theorem 6 on the suintervls [x i x i+ ], (i = n division d, we get (x i+ ) of the Z xi + x xi+ i+ x i ) f x i (x i+ x i ) mx f xi + x i+ jf (x i+)j + mx f xi + x i+ jf (x i )j Summing over i from to n nd tking into ccount tht jf j is usi-convex, we deduce tht M (f d) nx (x i+ x i ) mx f xi + x i+ jf (x i+)j i= + mx f xi + x i+ jf (x i )j which comletes the roof. Corollry 5. Let f I R! R e di erentile ming on I, I with <, nd f L[ ]. Given tht jf j is usi-convex on [ ], then in (3.), for every division d of [ ], () if jf j is incresing, then we hve (3.5) je (f d)j nx (x i+ x i ) f xi + x i+ + jf (x i+)j i= () if jf j is decresing, then we hve (3.6) je (f d)j nx (x i+ x i ) f xi + x i+ + jf (x i )j i= (3) if f xi+x i+ =, then we hve (3.7) je (f d)j nx (x i+ x i ) (jf (x i )j + jf (x i+ )j) i= () if f (x i ) = f (x i+ ) =, then we hve (3.) je (f d)j n X (x i+ x i ) f xi + x i+ i= Proof. The roof is similr to tht of Proosition, using Corollry. Proosition. Let f I R! R e di erentile ming on I, I with <. If jf j =( ) is usi-convex on [ ], >, then in (3.), for every
M. ALOMARI, M. DARUS, AND S.S. DRAGOMIR division d of [ ], the following holds " ( n X (3.9) je (f d)j (x ( + ) i+ x i ) mx f xi + x i+ i= ( jf o (x i+ )j + mx f xi + x i+ jf (x i )j Proof. The roof is similr to tht of Proosition, using Theorem. i= o Corollry 6. Let f I R! R e di erentile ming on I, I with <, nd f L[ ]. Given tht jf j is usi-convex on [ ], then in (3.), for every division d of [ ], () if jf j is incresing, then we hve n X je (f d)j (x ( + ) i+ x i ) f xi + x i+ + jf (x i+)j () if jf j is decresing, then we hve n X je (f d)j (x ( + ) i+ x i ) f xi + x i+ + jf (x i )j i= Proof. The roof is similr to tht of Proosition, using Corollry 3. Proosition 3. Let f I R! R e di erentile ming on I, I with <. If jf j is usi-convex on [ ],, then in (3.), for every division d of [ ], the following holds (3.) " je (f d)j nx (x i+ x i ) mx f xi + x i+ jf (x i+ )j i= ( )! 3 + mx f xi + x i+ jf (x i )j 5 Proof. The roof is similr to tht of Proosition, using Theorem 9. Corollry 7. Let f s in Proosition 3, if in ddition () jf j is incresing, then (3.5) holds. () jf j is decresing, then (3.6) holds. Proof. The roof is similr to tht of Proosition 3, using Corollry. References [] S.S. Drgomir, Two mings in connection to Hdmrd s ineulities, J. Mth. Anl. Al., 67 (99) 9 56. [] S.S. Drgomir nd R.P. Agrwl, Two ineulities for di erentile mings nd lictions to secil mens of rel numers nd to trezoidl formul, Al. Mth. Lett., (99) 9 95.
INEQUALITIES OF HERMITE-HADAMARD S TYPE [3] S.S. Drgomir, Y.J. Cho nd S.S. Kim, Ineulities of Hdmrd s tye for Lischitzin mings nd their licitions, J. Mth. Anl. Al., 5 (), 9 5. [] S.S. Drgomir nd S. Wng, A new ineulity of Ostrowski s tye in L norm nd lictions to some secil mens nd to some numericl udrture rule, Tmkng J. Mth., (997) 39. [5] S.S. Drgomir nd S. Wng, Alictions of Ostrowski s ineulity to the estimtion of error ounds for some secil mens nd for some numericl udrture rule, Al. Mth. lett., (99) 5 9. [6] D.A. Ion, Some estimtes on the Hermite-Hdmrd ineulity through usi-convex functions, Annls of University of Criov, Mth. Com. Sci. Ser., 3 (7), 7. [7] U.S. Kirmci, Ineulities for di erentile mings nd lictios to secil mens of rel numers to midoint formul, Al. Mth. Com., 7 (), 37 6. [] U.S. Kirmci nd M.E. Özdemir, On some ineulities for di erentile mings nd lictions to secil mens of rel numers nd to midoint formul, Al. Mth. Com., 53 (), 36 36. [9] M.E. Özdemir, A theorem on mings with ounded derivtives with lictions to udrture rules nd mens, Al. Mth. Com., 3 (3), 5 3. [] C.E.M. Perce nd J. Peµcrić, Ineulities for di erentile mings with liction to secil mens nd udrture formul, Al. Mth. Lett., 3 () 5 55. [] G.S. Yng, D.Y. Hwng nd K.L. Tseng, Some ineulities for di erentile convex nd concve mings, Com. Mth. Al., 7 (), 7 6. School Of Mthemticl Sciences, Universiti Kengsn Mlysi,, UKM, Bngi, 36, Selngor, Mlysi E-mil ddress mwomth@gmil.com E-mil ddress mslin@ukm.my E-mil ddress Sever.Drgomir@vu.edu.u School of Engineering & Science, Victori University,, PO Box, Melourne City, VIC, Austrli