Refinements to Hadamard s Inequality for Log-Convex Functions

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Alied Mhemics 899-93 doi:436/m7 Pulished Online Jul (h://wwwscirporg/journl/m) Refinemens o Hdmrd s Ineuli for Log-Convex Funcions Asrc Wdllh T Sulimn Dermen of Comuer Engineering College of Engineering Universi of Mosul Mosul Ir E-mil: wdsulimn@homilcom Received Aril 5 ; revised M 5 ; cceed M 8 In his er we show h log-convex funcion sisfies Hdmrd s ineuli s well s we give n exension for his resul in severl direcions Kewords: Log-Convex Funcions Hdmrd s Ineuli Inegrl Ineuli Inroducion Le f : I e convex ming of he inervl I of rel numers nd I wih < The following doule ineuli f f f f xdx () is known in he lierure s Hdmrd s ineuli In [] Fejer generlized he ineuli () roving h if g: is nonnegive inegrle nd smme ric o x nd if f is convex on [] hen f d d g x x f x g x x f f g x d x () A osiive funcion f is log-convex on rel inervl x we hve if for ll nd f x f x f (3) If he ove ineuli reversed hen f is ermed log-concve We define for x L x x x ln x ln x x In [] he following resul is chieved: Theorem Le f e osiive log-convex funcion on [ ] Then f d L f f (4) For f osiive log-concve funcion he ineuli is reversed Lemms The following lemms re needed for our im Lemm Le hen he following ineuli Proof Se We hve () f ln ln f ln ln for f Therefore f ins is minimum which is f Hence f which imlies e nd () follows Lemm For he following ineuli nd for () he following ineuli (3) Corigh SciRes

9 W T SULAIMAN we hve / Proof For which / imlies nd for which imlies We lso hve which imlies Se Then on keeing fixed we hve f for f f f f As f ins is minimum which is herefore nd (3) is sisfied Alhough some of he coming resuls (Lemm 3 nd heorem 3) re known u we rove hem new simle mehod Lemm 3 Le hen he following ineuli ln ln (4) Proof Lef ineuli Le us ssume h Se f x x x ln x x (5) 3 f x x x x s 4 34 x x x x 3 x Therefore f is non-decresing nd h imlies f x f The resul follows uing x in (5) Righ ineuli Le nd le x Se x f x ln x x (6) x We hve 4 4 fx s x x x x x Then f is non-decresing nd hence f x f The resul follows uing x in (6) Lemm 4 The funcion x f x x (7) ln x is non-decresing Proof x ln xx g x fx ln x ln gx x x herefore g is non-decresing Since g hen g x nd hence f x h is f is non-decresing 3 Theorems Theorem 3 Le f e osiive log-convex funcion on [] hen f sisfies () Proof This cn e chieved immediel s he log-convex funcion is convex which follows from he fc h Ever incresing convex funcion of convex ln f x funcion is convex which imlies h f x e is convex Or he roof cn e chieved following he definiion: Mking use of lemm we hve xx f f dx f xf xdx f xdx f xdx f xdx f d f d f f f f f f f f d d The following giving refinemen o heorem 3 Theorem 3 Le f e log convex funcion Then he following ineuli f f f f f f xdx f xdx ln f ln f (3) Proof Corigh SciRes

W T SULAIMAN 9 which imlies xx f f x f dx d / / ( )d d f x f x x f x x f x dx f xd x f f f f f f xdx f xdx ln f ln f in view of [] nd Lemm 3 The following resens n exension o Fejer s generlizion () for log-convex funcions Theorem 33 Le f e log convex g is osiive inegrle nd smmeric o x Then he following ineuli f gxdx f xgxdx f xgxdx f f f f g x x ln f ln g x x d d f (3) Proof xx / / f gxdx f gxd x f ( x) f x gxdx / / / / f x g ( x) f x g x dx f x g x dx f x g x dx f xgxdx f xgxdxdx f xgxdx which imlies Now for we hve f gxdx f xgxdx f xgxdx f x g x dx f g d f f g d f f f( ) f( ) gd gxd x ln f ln f in view of Lemms nd 3 Also we hve for d d f x g x x f g f f g d f f f f f gd gxd x ln f ln Corigh SciRes

9 W T SULAIMAN in view of Lemms nd 3 Conseuenl we oin Lemm f f f f d d ln f ln f f x g x x g x x g x d x This comlees he roof of he heorem Theorem 34 Assume h f : I e n incresing log-convex funcion Then for ll we hve The following is noher refinemen of heorem 3 f f f f f w f xdx W (33) ln f ln f where Proof We hve vi Lemms 3 nd 4 ( ) 33 w f f 4 4 (34) f f f f W ln f ln f ln f ln f 3 333 f f f f w() f xd d 4 4 4 4 x f x x ( ) f xdx f xdx f xd x f( x)dx f xdx ( ) ( ) ( ) f( x) dx f xdx ( ) ( ) f f( ) f f ( ) W( ) ln f ln f( ) ln f ln f f f f f f f f f ( ) ln f ln f( ) ln f ln f ln f ln f Theorem 35 Le f is log-convex nd g is hen non-negive inegrle Proof We hve vi Holder s ineuli he following ineuli (35) f f f xgxdx g xdx ln f ln f (36) f x g x dx f x dx g x dx f d g x dx ( ) f f f dg xdx f d g ( x)dx f f f g ( x)dx ln f ln f Corigh SciRes

4 References W T SULAIMAN 93 369-39 [] P M Gill C E M Perse nd J Pečrić Hdmrd s [] L Fejér Uer die Fourierreihen II Mh Nurwiss Anz Ungr Akd Wiss Hungrin Vol 4 96 Ineuli for R-Convex Funcions Mhemicl Ineuliies & Alicions Vol 5 No 997 46-47 Corigh SciRes

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