SEVERAL GEOMETRIC INEQUALITIES OF ERDÖS - MORDELL TYPE IN THE CONVEX POLYGON

Size: px

Start display at page:

Download "SEVERAL GEOMETRIC INEQUALITIES OF ERDÖS - MORDELL TYPE IN THE CONVEX POLYGON"

Shanon Hutchinson
5 years ago
Views:

1 INTERNATIONAL JOURNAL OF GEOMETRY Vol. 1 (01), No. 1, 0-6 SEVERAL GEOMETRIC INEQUALITIES OF ERDÖS - MORDELL TYPE IN THE CONVEX POLYGON NICUŞOR MINCULETE Abstract. I this aer we reset the several geometric iequalities of Erdös-Mordell tye i the covex olygo, usig the Cauchy Iequality. 1. Itroductio I [6], i colaboratio with A. Gobej, we reset some geometric iequalities of Erdös-Mordell tye i the covex olygo. Here, we foud others geometric iequalities of Erdös-Mordell tye, usig several kow iequalities, i the covex olygo. Let A 1,A ;...,A the vertices of the covex olygo, 3; ad M, a oit iterior to the olygo. We ote with the distaces from M to the vertices A k ad we ote with the distaces from M to the sides [A k A k+1 ] of legth A k A k+1 = a k ; where k = 1; ad A +1 A 1. For all k f1; :::; g with A +1 A 1 ad m A k\ MA k+1 = k we have the followig roerty: ::: + = : L. Fejes Tóth cojectured a iequality which is refered to the covex olygo, recall i [1] şi [3], thus (1) cos : I 1961 H.-C. Lehard roof the iequality (1), used the iequality () w k cos ; which was established i [5], where w k the legth of the bisector of the agle A k MA k+1, (8) k = 1; with A +1 A 1. Keywords ad hrases: Iequality of Erdös-Mordell tye, Cauchy s Iequality (010) Mathematics Subject Classi catio: 51 Mxx,6D15

2 Several geometric iequalities of Erdös-Mordell tye i the covex olygo 1 M. Dic¼a ublished other solutio for iequality (1) i Gazeta Matematic¼a Seria B i 1998 (see [4]). Aother iequality of Erdös-Mordell tye for covex olygo was give by N. Ozeki [9] i 1957, amely, (3) Y sec Y w k which roved the iequality 16.8 from [3] due to L. Fejes-Tóth, so Y (4) sec Y : R. R. Jaić i [3], shows that i ay covex olygo A 1 A ::: A, there is the iequality (5) si A k : D. Buşeag roosed i GMB o. 1/1971 the roblem 10876, which is a iequality of Erdös-Mordell tye for covex olygo, thus, a k (6) ; where is the semierimeter of olygo A 1 A ::: A ad is the area of olygo. I coectio with iequality (6), D. M. B¼atieţu established [] the iequality (7) a k r if the olygo A 1 A ::: A is circumscribed about a circle of radius r. Amog the relatios established betwee the elemets of olygo A 1 A ::: A we ca remark the followig relatio for - the area of covex olygo A 1 A ::: A : (8) = a 1 r 1 + a r + ::: + a r : We select several iequalities obtaied from [6]: (9) 1 + si A k hold for all k f1; ; :::; g ; with r 0 = r ; (10) cos Y Y ( 1 + ) ; (r 0 = r ) ad (11) ad (1) R k 1 + sec cos :

3 Nicuşor Miculete. Mai results First, we will follow some rocedures used i aer [6], through which we will obtai some Erdös-Mordell-tye iequalities for the covex olygo. Amog these will aly the Cauchy Iequality Theorem.1. I ay covex olygo A 1 A ::: A, there is the iequality (13) cos + +1 : Proof. The iequality (11), 1 + cos ; with r 0 = r ; is exaded i the followig way, r + r 1 + r 1 + r + ::: + r 1 + r = R 1 R R 1 r r ::: + r + 1 cos R 1 R R R 3 R R 1 O the other had, we have ; (8) k = 1; with R 0 = R ; from where we ca deduce aother iequality, of a Erdös- Mordell tye, amely, + +1 cos : Theorem.. For ay covex olygo A 1 A ::: A,we have the iequality + +1 cos ; with R +1 = R 1 : Proof. The iequality y k y k! is well kow, because it is a articulary case of Cauchy s iequality. I this we will take = ad y k = 1. Thus, the iequality becomes

4 Several geometric iequalities of Erdös-Mordell tye i the covex olygo 3 ad, if we use the iequality (13), we get + +1 cos ; with R +1 = R 1 : Theorem.3. I ay covex olygo A 1 A ::: A, there is the iequality rk 1 (15) cos : Proof. From Cauchy s iequality, we have y k y k Usig the substitutios = rk 1! : ad y k = 1, we deduce that the iequality rk 1 ; rk 1 holds. However, from the relatio (11), we obtai rk 1 cos which imlies the iequality rk 1 cos ; with r 0 = r : Remark 1. The iequality (15) geeralizes the roblem 1045 of G. Tsisifas from the magazie Crux Mathematicorum. This is also remarked i [7]. Theorem.4. I ay covex olygo A 1 A ::: A (16) with r 0 = r : Proof. I the iequality y k we relaced = y k = Rk rk 1 cos ; y k! rk 1 ad the iequality becomes there is the iequality R k rk 1! rk 1 cos :

5 4 Nicuşor Miculete This meas that iequality (16) is true. Theorem.5. I ay covex olygo A 1 A ::: A there is the iequality with A +1 = A 1 : a k +1 si ; Proof. We ca be writte the area of triagle A k MA k+1 i two ways, thus a k which imlies the relatio = +1 si A k MA k+1 ; a k +1 = si A k MA k+1 so, by assig to the sum, we get the relatio a k = si A k MA k+1 ; +1 with A +1 = A 1 : Because the fuctio f : (0; 1)! R, is de ed as f (x) = si x, is cocave, we will aly the iequality Jese, thus! 1 si A k MA k+1 si 1 A k MA k+1 = si : Therefore, we have A k MA k+1 si : Cosequetly, we obtai the iequality of the statemet. Remark. The equality hold i the above metioed theorems whe the olygo is regular. Remark 3. O the a had, we have the equality (8), = a 1 r 1 + a r + ::: + a r = a k ; ad o the other had, we have the iequality Cauchy, where we will relace = q a k ad y k = ak,the a k = a k which roves the iequality (6): a k! a k = 4 Theorem.6. I ay covex olygo A 1 A ::: A there is the iequality (18) Rk si A k sec! holds.

6 Several geometric iequalities of Erdös-Mordell tye i the covex olygo 5 Proof. Alyig iequality (9), we have the iequality = MA k 1 + si A (8) k = 1; k with r 0 = r ; ad this, by squarig, becomes 4Rk si A k ( 1 + ) si A k ad takig the sum, we deduce " # ( 1 + ) 4 R k si A k so, we foud iequality (18). ( 1 + ) si A k si A k 4! cos Theorem.7. I ay covex olygo A 1 A ::: A there is the iequality (19) ( + +1 ) sec! holds. Proof. From iequality (9), we have = MA k 1 + si A ; (8) k = 1; k with r 0 = r ; ad this, by multily with ( 1 + ),becomes ( 1 + ) ( 1 + ) si A k ad by assig to the sum, we obtai the relatio " ( 1 + ) ( 1 + ) ( 1 + ) si A k Therefore, we have ( 1 + ) sec But, it follows that ( + +1 ) = # si A k! : ( 1 + ) which meas that, we obtai the iequality of statemet. 4! cos :

7 6 Nicuşor Miculete Refereces [1] Abi-Khuzam, F. F., A Trigoometric Iequality ad its Geometric Alicatios, Mathematical Iequalities & Alicatios, 3(000), [] B¼atieţu, D. M., A Iequality Betwee Weighted Average ad Alicatios (Romaia), Gazeta Matematic¼a seria B, 7(198). [3] Botema, O., Djordjević R. Z., Jaić, R. R., Mitriović, D. S. ad Vasić, P. M., Geometric Iequalities, Groige,1969. [4] Dic¼a, M., Geeralizatio of Erdös-Mordell Iequality (Romaia), Gazeta Matematic¼a seria B, 7-8(1998). [5] Lehard, H.-C., Verallgemeierug ud Verscharfug der Erdös-Mordellsche Ugleichug für Polygoe, Arch. Math., 1(1961), [6] Miculete, N. ad Gobej, A.,Geometric Iequalities of Erdös-Mordell Tye i a Covex Polygo (Romaia), Gazeta Matematic¼a Seria A, r. 1-(010). [7] Miculete, N., Geometry Theorems ad Seci c Problems (Romaia), Editura Eurocaratica, Sfâtu Gheorghe, 007. [8] Mitriović, D. S., Peµcarić, J. E. ad Voleec, V., Recet Advaces i Geometric Iequalities, Kluwer Academic Publishers, Dordrecht, [9] Ozeki, N., O P. Erdös Iequality for the Triagle, J. College Arts Sci. Chiba Uiv., (1957), [10] Vod¼a, V. Gh., Sell Old Geometry (Romaia), Editura Albatros,1983. "DIMITRIE CANTEMIR" UNIVERSITY 107 BISERICII ROMÂNE, BRAŞOV, ROMANIA address: miculete@yahoo.com

A REFINEMENT OF JENSEN S INEQUALITY WITH APPLICATIONS. S. S. Dragomir 1. INTRODUCTION

TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 1,. 153-164, February 2010 This aer is available olie at htt://www.tjm.sysu.edu.tw/ A REFINEMENT OF JENSEN S INEQUALITY WITH APPLICATIONS FOR f-divergence