DSpace at VNU: One result of the cyclic inequality

V N U JO U R N A L OF SCIENCE, M a th e m a tics - Physics T X X II, N q - 2006 ONE RESULT OF THE CYCLIC INEQUALITY N guyen Vu Luong D epartm ent o f M athem atics - Mechanics - Informatics College o f Science , V N U A bstract paring with In this paper we present some inequalities which are obtained from com­ n~ỹ -pOL Q S(a, Ị3) = V - - llz l—_ J - Xn k = l (Tfc+ + x k+2)p (Xn + Xi )0 (xi + X2)P and *(«.£) = ả w here a e n + , € R + ,X i € /Ỉ+ k= (i = ĩ~ỉí), R+ = {x e R\x > 0} I I n tr o d u c tio n Some cyclic inequalities have presented under simple forms but they are re a lly difficult to prove he Shapiro’s inequality is a very special inequality and it is suprising th at many m athem aticians have spent time on it W hen a = p = we obtain the Shapiro’s inequality (1 ,1 ) ^ R ( l , 1) ( n > 3) (1.1) This inequality is correct for odd intergers less than or equal to 23 and for even mtergers less than or equal to 12 For all other n, the inequality is false For T h e o r e m 1.1 If Xi Ễ i?+ (i = l , n ) , Q > l , n is an odd integer less than or equal to 23 and an even integer less than or equal to 12, then S( a, a ) > R ( a a) (1.2) Equality o f (1.2) holds i f and only i f X = x = ■■■= x n Typeset by 39 N g u y e n Vu L u o n g 40 Proof Using the inequality n by (1.1) we have I I I C a se a < p In this case we obtain result T h e o r e m 1.2 I f a & R +, (3 G i ỉ + , a < p then (i) The inequality S( a , P) ^ R(a,(3) is not true for every positive Xi (i = 1, n) (ii) The inequality S( a, /3) ^ R(a,(3) is not true for every positive Xi (i = 1,n ) Proof, (i) Taking Xi = 1, X2 = X3 = = I n = a > we obtain m S {a ,0 ) r1 [4 2^ - A + n —3, a ^ -“ J 2aa (1 + a)^3’ 773— ] + = ^ g [l + ( n - l ) a “ - /ĩ] Since /3 > a , it follows lim (a ,/J ) = a —► 4-00 lirn R ( a , ) = l g a —>-f oc For large enough a in (a ,/? ) and i?(a,/3), we have S(a,/3) < i? ( a ,/3) It follows th at S(a,/3) ^ R (a,/3) is wrong (u) Taking Xi = a, X2 = £3 = = x n = 1(a > 0) we obtain _, m aa n - + (n - !)]• Since (3 > a , it follows lim (a,/? ) = + 00 —►-foo 71 — lim i? (a ,/3) a—>+ 00 ?2^ For large enough a in (a,/3 ) and i?(a,/?), we have S (a,/3) > /ỉ(a ,/3 ) It follows th at 5(a,/3) ^ R ( a , P) is wrong O ne R e su lt o f the C y clic In eq u a lity 41 IV C a se a > /3 Given T h e o re m 1.3 Xi (i = 1, n) are positive numbers , p, q are positive integer numbers such that p > q We obtain Jlf — ill 4_ , _ I n-1 , xn > (x2 4-X3)9 O + Z4)9 (Xn+Xi)*? ( £ i + £ 2)9 29 Proof Lets consider the case p-q , _ I Tp- 29+ x£ + (an + x 2)9x r 2? ^ 2^+1x r Summing all above inequalities, we have 22qM + x r 2> + Z3 )9 + • • • + XP-29( X! + X2)9 Ỉ? 2«+ i ( x r + x r + • • • + x r 9)- ( ) Moreover, X1 _2