Journal of Inequalities in Pure and Applied Mathematics

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1 Journl of Inequlities in Pure nd Applied Mthemtics ON LANDAU TYPE INEQUALITIES FOR FUNCTIONS WIT ÖLDER CONTINUOUS DERIVATIVES LJ. MARANGUNIĆ AND J. PEČARIĆ Deprtment of Applied Mthemtics Fculty of Electricl Engineering nd Computing University of Zgreb Unsk 3, Zgreb, Croti. EMil: Fculty of Tetile Technology University of Zgreb Pierottijev 6, Zgreb Croti. EMil: volume 5, issue 3, rticle 72, Received 08 Mrch, 2004; ccepted April, Communicted by: N. Elezović Abstrct ome Pge c 2000 Victori University ISSN (electronic):

2 Abstrct An inequlity of Lndu type for functions whose derivtives stisfy ölder s condition is studied Mthemtics Subject Clssifiction: 26D5 Key words: Lndu inequlity, ölder continuity Introduction Min Results References Pge 2 of

3 . Introduction S.S. Drgomir nd C.I. Pred hve proved the following theorem (see []): Theorem A. Let I be n intervl in R nd f : I R loclly bsolutely continuous function on I. If f L (I) nd the derivtive f : I R stisfies ölder s condition (.) f (t) f (s) t s for ny t, s I, where > 0 nd (0, ] re given, then f L (I) nd one hs the inequlities: (.2) f [ ( )] f f m(i) + if m(i) (+) [m(i)] ( f if 0 < m(i) ) + ( + ) + ; ( ) f + ( + ) +, where is the -norm on the intervl I, nd m(i) is the length of I. In our pper we shll give n improvement of this theorem. Pge 3 of

4 2. Min Results Theorem 2.. Let I be n intervl nd f : I R function on I stisfying conditions of Theorem A. Then f L (I) nd the following inequlities hold: (2.) f [ ( )] f + + if m(i) f m(i) + + [m(i)] if 0 < m(i) 2 + ( ) f + ( + ( ) f ) + ; + ( + ) +, where is the -norm on the intervl I, nd m(i) is the length of I. In our proof nd in the subsequent discussion we use three lemms. Lemm 2.2. Let, b R, < b, (0, ]. Then the following inequlity holds: (2.2) (b ) + + ( ) + (b ) +, [, b]. Proof. Consider the function y : [, b] R given by: y() = (b ) + + ( ) +. We observe tht the unique solution of the eqution y () = ( + ) [( ) (b ) ] = 0 Pge 4 of

5 is 0 = +b 2 [, b]. The function y () is decresing on (, 0 ) nd incresing on ( 0, b). Thus, the miml vlues for y() re ttined on the boundry of [, b] : y() = y(b) = (b ) +, which proves the lemm. A generliztion of the following lemm is proved in []: Lemm 2.3. Let A, B > 0 nd (0, ]. Consider the function g : (0, ) R given by: (2.3) g (λ) = A λ + B λ. Define λ 0 := ( ) A + (0, ). Then for λ B (0, ) we hve the bound (2.4) inf λ (0,λ ] g (λ) = Proof. We hve: A λ + B λ if 0 < λ < λ 0 ( + ) + A + B + if λ λ 0. g (λ) = A λ 2 + B λ. The unique solution of the eqution g (λ) = 0, λ (0, ), is λ 0 = ( ) A + B (0, ). The function g (λ) is decresing on (0, λ 0 ) nd incresing on (λ 0, ). The globl minimum for g (λ) on (0, ) is: (2.5) g (λ 0 ) = A which proves (2.4). ( ) ( ) B + A + + B = ( + ) + A + B +, A B Pge 5 of

6 Lemm 2.4. Let A, B > 0 nd (0, ]. Consider the functions g : (0, ) R nd h : (0, ) R defined by: (2.6) g (λ) = A + B λ λ h (λ) = 2A + B λ. λ 2 Define λ 0 := ( ) A + (0, ). Then for λ B (0, ) we hve: (2.7) inf g (λ) < λ (0,λ ] inf λ (0,λ ] h (λ) if 0 < λ < 2λ 0 inf g (λ) = inf h (λ) if λ 2λ 0. λ (0,λ ] λ (0,λ ] Proof. In Lemm 2.3, we found tht the globl minimum for g (λ) is obtined for λ = λ 0. Similrly we find tht the globl minimum for h (λ) is obtined for λ = 2λ 0, nd its vlue is equl to the miniml vlue of g (λ), i.e. h (2λ 0 ) = g (λ 0 ). The only solution of eqution g (λ) = h (λ), λ (0, ), is: [ λ S = A B( 2 ) ] +, nd we cn esily check tht λ 0 < λ S < 2λ 0. Thus, for λ < λ 0 we hve g (λ ) < h (λ ) nd inf g (λ) < inf h (λ), nd the rest of the proof is λ (0,λ ] λ (0,λ ] obvious. Pge 6 of

7 Proof of Theorem 2.. Now we strt proving our theorem using the identity: (2.8) f() = f() + ( )f () + or, by chnging with nd with : [f (s) f ()]ds;, I (2.9) f() = f() + ( )f () + Anlogously, we hve for b I: (2.0) f(b) = f() + (b )f () + From (2.9) nd (2.0) we obtin: (2.) f(b) f() = (b )f () + nd (2.2) f () = f(b) f() b + b [f (s) f ()]ds;, I. [f (s) f ()]ds; b, I. [f (s) f ()]ds [f (s) f ()]ds;, b, I [f (s) f ()]ds b [f (s) f ()]ds. Pge 7 of

8 Assuming tht b > we hve the inequlity: (2.3) f f(b) f() () + b b + b Since f is of ölder type, then: b f (s) f () ds (2.4) s ds (2.5) = f (s) f () ds f (s) f () ds. (s ) ds = + (b )+ ; f (s) f () ds s ds = From (2.3), (2.4) nd (2.5) we deduce: (2.6) f () f(b) f() b ( s) ds b, I, b > = + ( )+ ;, I, <. Pge 8 of

9 + (b )( + ) [(b )+ + ( ) + ];, b, I, < < b. Since f L (I) then f(b) f() 2 f. Using Lemm 2.2 we obviously get tht: (2.7) f () 2 f b + + (b ) ;, b, I, < < b. Denote b = λ. Since, b I, b >, we hve λ (0, m(i)), nd we cn nlyze the right-hnd side of the inequlity (2.7) s function of vrible λ. Thus we obtin: (2.8) f () 2 f λ + + λ = g (λ) for I nd for every λ (0, m(i)). Tking the infimum over λ (0, m(i)) in (2.8), we get: (2.9) f () inf g (λ). λ (0,m(I)) If we tke the supremum over I in (2.9) we conclude tht (2.20) sup f () = f inf g (λ). I λ (0,m(I)) Mking use of Lemm 2.3 we obtin the desired result (2.). Pge 9 of

10 Remrk 2.. Denote λ 0 = [ 2 ( + ) ] f +. Compring the results of Theo- rem A nd Theorem 2. we cn see tht in the cse of m(i) 2λ 0 the estimted vlues for f in both theorems coincide. If 0 < m(i) < 2λ 0 the estimted vlue for f given by (2.) is better thn the one given by (.2). Nmely, using Lemm 2.4 we hve: (2.2) 2 f m(i) + + [m(i)] < 4 f m(i) + 2 ( + ) [m(i)] ; m(i) (0, λ 0 ] nd (2.22) [ ( 2 + )] + f + + < 4 f m(i) + 2 ( + ) [m(i)] ; m(i) [λ 0, 2λ 0 ). Remrk 2.2. Let the conditions of Theorem 2. be fulfilled. Then simple consequence of (2.) is the following inequlity: (b )f () f(b) + f() [ (b ) + + ( ) +] ; +, b, I, < < b. This result is n etension of the result obtined by V.G. Avkumović nd S. Aljnčić in [2] (see lso [3]). Pge 0 of

11 References [] S.S. DRAGOMIR AND C.J. PREDA, Some Lndu type inequlities for functions whose derivtives re ölder continuous, RGMIA Res. Rep. Coll., 6(2) (2003), Article 3. ONLINE [ html]. [2] V.G. AVAKUMOVIĆ AND S. ALJANČIĆ, Sur l meilleure limite de l dérivée d une function ssujetie à des conditions supplementires, Acd. Serbe Sci. Publ. Inst. Mth., 3 (950), [3] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, Dordrecht, Boston, London, 99. Pge of

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