EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Sayı: 3-1 Yıl:

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1 EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-9 NEW INEQUALITIES FOR CONVEX FUNCTIONS KONVEKS FONKSİYONLAR İÇİN YENİ EŞİTSİZLİKLER Mevlüt TUNÇ * ve S. Uğur KIRMACI Kilis 7 Arlık Üniversitesi, Fen Edeiyt Fkültesi Mtemtik Bölümü, Kilis Attürk Üniversitesi, KKEF, Mtemtik Eğitimi A.B.D., 54, Erzurum ABSTRACT Geliş Trihi: 8 Nisn Kul Trihi: 5 Myıs In the present pper we estlish some new integrl inequlities nlogous to the well known Hdmrd's inequlity y using firly elementry nlysis. ÖZET Bu mklede iz temel nliz işlemlerini kullnrk litertürde iyi ilinen Hdmrd eşitsizliğine enzer yeni integrl eşitsizlikleri kurduk. Key Words: Hdmrd's inequlity, convex function, concve function, specil mens..introduction The following inequlity [see Drgomir, (99)] f f f f ( x) dx (.) Which holds for ll convex functions f :[, ] is known in the literture s Hdmrd's inequlity. Since its discovery in 893, Hdmrd's inequlity [see Hdmrd, (893)] hs een proven to e one of the most useful inequlities in mthemticl nlysis. A numer of ppers hve een written on this inequlity providing new proofs, noteworthy extensions, generliztions nd numerous * Sorumlu Yzr: mevluttunc@kilis.edu.tr

2 9 Tunç & Kırmcı pplictions, see the references cited therein. The min purpose of this pper is to estlish some new integrl inequlities nlogous to tht of Hdmrd's inequlity given in (.) involving two convex functions. The nlysis used in the proof is elementry.. MAIN RESULT We need the following Lemm proved in [see, Pecric nd Drgomir, 4pp, (99) ] which dels with the simple chrcteriztion of convex functions. Lemm A: The following sttements re equivlent to mpping: f :[, ] ; i) f is convex on [, ], ] :[,] ii) for ll x, y in [, the mpping g t f ( tx t y = ( ) ) is convex on [ ],. g, defined y For the proof of this Lemm, see [Pecric nd Drgomir, 4pp, (99)]. Our min result is given in the following theorem. Theorem.: Let f :[, ] e nonnegtive nd convex function. Then one hs the inequlity: ( ) ( ) (, ) f f ( x) f ( x) dx ( x ) f ( x) dx f x dx M 3 (.) where M (, ) = f f f f. Proof: Since f is convex function on [, ], then we hve tht f ( t ( t) ) tf ( t) f EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

3 New Inequlıtıes For Convex Functıons 93 for ll t [,]. Using the elementry inequlity G, A,,,, we cn conclude tht { tf f ( t ( t) ) ( t) f f ( t ( t) ) } f ( t ( t) ) t f t( t) f f ( t) f By the Lemm A f ( t ( t) ) is convex on [, ], it is integrle on [, ]. Integrting oth sides of the ove inequlity over t on [, ] we get ( ( ) ) ( ) ( ( ) ) f tf t t dt f t f t t dt ( ( ) ) f t t dt f t dt f f t t dt f t dt By sustituting ( ) ( ( ) ) tf t x = t t, it is esy to oserve tht x = f ( x ) dx = ( x ) f ( x ) dx nd t dt ( ) ( ( ) ) t f t t dt x = f ( x ) dx = ( x ) f ( x ) dx EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

4 94 Tunç & Kırmcı It cn e esily checked tht tdt= ( t) dt=, t( t) dt= 3 6 ( ( ) ) f t t dt = f x dx When ove equlities re tken into ccount, the proof is complete. Teorem.: Let f :[, ] e nonnegtive nd convex function. Then one hs the inequlity: f ( x) dx f f x dx (, ) N 4 f ( ) 4 f N, = f 4f f f. where Proof: Since f is convex function on [, ], then we hve tht t ( t) ( t) t f = f ( ( ) ) (( ) ) (.) f t t f t t t. Similrly s explined in the proof of inequlity (.) for ll [,] given ove, we otin EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

5 New Inequlıtıes For Convex Functıons 95 f f t t f f t t f ( ( ) ) (( ) ) ( ( ) ) f t t 4 ( ( ) ) (( ) ) (( ) ) f t t f t t f t t 4 f ( ( ) ) (( ) ) ( ) ( ) = ( ) f t t f t t 4 tf t f t f tf f ( ( ) ) (( ) ) f t t f t t 4 t t f f t t f f Likewise s explined in the proof of inequlity (.) given ove, we integrte oth sides of the ove inequlity over on,, we get t [ ] f f ( t ( t) ) dt f f (( t) t) dt { ( ( ) ) (( ) )} f dt f t t f t t dt 4 ( ) f f f f ( ) t t dt t t dt By sustituting t ( t) = x nd ( t) t= y it cn e esily otined, f ( t ( t) ) dt = f (( t) t) dt = f ( x) dx t ( t) dt =, t ( t) dt 6 = 3 EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

6 96 Tunç & Kırmcı From the ove equlities it is esily otined tht f ( x) dx f ( x) dx = f ( x) dx f f f ( ) f f f f f f x dx 3 Now dividing oth sides of the ove inequlity y f we get the desired inequlity in (.). The proof is complete. Remrk: If we choose = nd = nd the convex functions f x = x, then it is esy to oserve tht the inequlities otined (.) nd (.) re certin in the sense tht we hold equlities in (.) nd (.). Theorem.3: Let f :[, ] e nonnegtive nd convex function. Then ( ) ( ( ) )( ( ) ) f tx t y tf x t f y dtdydx ( ) f tx t y dtdydx ψ (, ) f ( x) dx 3( ) where ψ (, ) = f f f f. Proof: Since f is convex function on [, ], then we hve tht f ( tx ( t) y) tf ( x) ( t) f ( y) for ll x, y [, ] nd t [,] G(, ) A (, ),,, we get, (.3). Using the elementry inequlity EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

7 New Inequlıtıes For Convex Functıons 97 ( ( ) )( ( ) ) ( ( ) ) f tx t y tf x t f y f tx t y t f x t( t) f ( x) f ( y) ( t ) f ( y) As explined in the proof of inequlity (.) given ove, we integrte oth sides of tht ove inequlity over t on [, ], nd we otin ( ( ) )( ( ) ) f tx t y tf x t f y dt ( ( ) ) f tx t y dt f x t dt f x f y t t dt f y t dt = ( ( ) ) f tx t y dt f x f x f y f y (.3.) Integrting oth sides of tht ove inequlity (.3.) over x nd on [, ] we otin ( ) ( ( ) )( ( ) ) f tx t y tf x t f y dtdydx ( ( ) ) f tx t y dtdydx 3 f x dydx f x f y dydx f y dydx 3 3 ( ( ) ) f tx t y dtdydx f ( x) dx f ( x) dx f ( y) dy 3 3 y (.3.) By using the right hlf of the Hdmrd's inequlity given in (.) on the right side of (.3.) we write EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

8 98 Tunç & Kırmcı ( ( ) )( ( ) ) f tx t y tf x t f y dtdydx ( ) f ( tx ( t) y) dtdydx f ( x) dx 3 f f f f 3 ( ) = f ( tx ( t) y) dtdydx f ( x) dx 3 ( f f f f )( ) Now multiplying oth sides of the ove inequlity y ( ) we get desired inequlity in (.3). The proof is complete. Theorem.4: Let f :[, ] e nonnegtive nd convex function. Then f tx ( t) tf ( x) ( t) f dtdx (, ) ( ) ψ f tx ( t) dtdx where ψ (, ) = f f f f. Proof: Since f is convex function on [, ], then we hve tht f tx ( t) tf ( x) ( t) f x, y, t,, we get for ll [ ] nd [ ] f tx ( t) ( ) tf x t f f tx ( t) t f ( x) t( t) f ( x) f ( t) f (.4) EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

9 New Inequlıtıes For Convex Functıons 99 As explined in the proof of inequlity (.) given ove, we integrte oth sides of tht ove inequlity over t on [, ], nd we otin f tx ( t) tf ( x) ( t) f dt ( ) f tx t dt f x t dt f f ( x) t( t) dt f ( t) dt f tx t dt f x f f x f (.4.) = ( ) As explined in the proof of inequlity (.3) given ove, integrting,, nd using oth sides of tht ove inequlity (.4.) over x on [ ] the right hlf of the Hdmrd's inequlity given in (.) nd convexity of f we otin f tx ( t) tf ( x) ( t) f dtdx ( ) f tx t dtdx ( ) ( ) f ( x) dx f f ( x) dx f f tx t dtdx f f ( ) 3 f f f f f f 3 3 = ( ) f tx t dtdx (, ) ( ) ( ) ( ) ψ (.4.) EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

10 Tunç & Kırmcı Now dividing oth sides of (.4.) y ( ) nd rewriting (.4.) we get the required inequlity in (.4). The proof is complete. 3. APPLICATIONS FOR SPECIAL MEANS As in [Kırmcı, 4], we shll consider the mens s ritrry rel numers,,. In the resources there includes A (, ) =,, (rithmetic men) K(, ) =,, (qudrtic men) L(, ) =,,,, (logrithmic ln ln men) G, =,, (geometric men). Now, using the results of Section, we illustrte some pplictions of specil mens of rel numers. Proposition 3.: Let < <. Then one hs the inequlity, 4 A(, ) K (, ) L, 3 G, 3 Proof: The proof is immedite from Theorem. s pplied for f ( x) = nd the detils re omitted. x Proposition 3.: Let < <. Then one hs the inequlity, A(, ) A(, )( K (, ) 4 G (, )) L, A, 4 G, 4 G, Proof: The ssertion follows from Theorem. s pplied to f ( x) = nd the detils re omitted. x EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-

11 New Inequlıtıes For Convex Functıons REFERENCES Drgomir, S.S. (99) Two mppings in connection to Hdmrd's inequlities, J.Mth.Anl.Appl. 67, Hdmrd, J. (893).Etude sur les properties des fonctions entieres et en prticulier d'une fonction consideree pr Riemnn, J.Mth.Pures ppl. 58, 7-5. Heing, H.P., Mligrnd, L. (99/9) Cheyshev inequlity in function spces, Rel Anlysis Exchnge 7, -47. Kırmcı, U.S. (4). Inequlities for differentile mppings nd pplictions to specil mens of rel numers nd to midpoint formul, Appl. Mth.Comput. 47, Mligrnd, L., Pecric, J.E. nd Persson, L.E. (994). On some inequlities of the Gruss-Brnes nd Borell type, J.Mth.An.Appl. 87, Mitrinovic, D.S. Anlytic Inequlities, Springer-Verlg, Berlin, New York 97. Pchptte, B.G. (3). On some inequlities for convex functions, RGMIA Res. Rep. Coll., 6(E) Pecric, J.E., Drgomir, S.S. (99). A generliztion of Hdmrd's inequlity for Isotonic liner functionls, Rdovi Mtemticki 7, 3-7. **** EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-