VNU JOURNAL OF SCIENCE, M athem atics - Physics, T X X II N q - 2006 S H A R P E N I N G M E A N -F O R M A N D C Y C L IC IN E Q U A L IT Y N guyen Vu Luong Department o f M athem atics - Mechanics - Informatics, College o f Science, VNƯ A bstract In this paper, we construct some sharpening forms of the mean form and sharpening some types of cyclic inequalities I n t r o d u c t i o n In the international conference 1996, Zivojin Mijakovic and Milan Mijakovic pre sented a method to sharpen AM-GM inequality in their paper Starting with the AM-GM inequality where (lị (i = l , n ) are positive numbers, they created some stronger inequalities where is a non-decreasing monotonic function By virtur of the results, they created many infinitely symmetric expressions, th a t depend on parameter a , between G„(a) and i4n (a) In this paper, we will sharpen some types of the inequalities with r > 1, dfc > 0(k = 1, n) 1) and 2) with < r < 1, (k = 1, n) For p > q > 1, we have with a* > (k = 1, n) Typeset by 21 N g u y e n Vu Luong 22 3) V -— -+ — —— J ^ a fc + a fc+1 a„ + n —2 Y ^ 4) ’ (where with ajfc > 0(fc= l , n ) „2 „2 n _Ạ + » 1— + — ^ > - i - y a k ak+l + (3ak+2 an + /3a Oi + a + /? " ajt > 0, /c = ITn and /? > is given) S h a r p e n i n g m e a n - f o r m in e q u a lity We denote Bn (a ,p ,q ,a )= where a A: > n £ ^ { a pk + a ) ? ) p - a 9, k=l (k = 1, n );p > 0; Ợ > 0; a > We have £ n (a,p, 1, a ) = ( - + a ) p) p - Q k= 1 n B „ (a ,p ,1 ,0 )= ( £ l > ỉ ) fc= l V - B n (a, , , ) = — 2_J a k- fc=i Let us consider the inequality B n (a, r, 1,0) ^ B n (a, ,1 ,0), wherer > B n (a, r, 1,0) ^ B n (a, , 1, 0)where0 < r < L e m m a 1.1 z?„(a,r, 1, a ) is a non - increasing monotonic function (with variable a) Proof We have Z?;(a,r,l,Q)= Z(afc + a)r)^ fc=i Therefore Y , r(a* + Q)r_1)l fc=i , p/ / \ ỉE ìĩĩC a ib + a r B n {a,r, 1,a ) = — — — - EZT - L ( i E S V + a r) ; ~ L S h arpenin g m e a n - f o r m a n d cyc lic inequality Denote Ak = (a* + a ) r_1 and q = We get J_ ir^k—n r ,l,a ) = t ,, - (ÌE Ỉ:^ ỉ) A If r > 1, then Ợ > and if < r < 1, then + « ) ') *=1 * * (- ; fc=i + a ) r ) 1/r - a > fc=i A:=l + « ) ) - < * fc=i + ^ n ^ (afc+a) Jt=l For < r < 1, the proof is similar For p > q > 1, we have B n (a,p, 1,0) ^ n (a,ợ, 1,0) Theorem 1.1 Given p > q > 1, a ^ 0, Ojfc > 0, (k = T~ri) Then ( i) B n (a,p, 1,0) ^ B n (a, p, q, a) ^ B n (a,q, 1,0) ( ii) B n (a,p, q, q ) is a non-increasing monotonic function (in variable a) N g u y e n Vu L u on g 24 Proof (i) Denote Ak — aqk , the required inequality is equivalent to !/2 ặ (a + ajt-fi)2 Hence G n (a, a) is a non-increasing monotonic function Since Q ^ 0, we get G n (a ,a ) ^ G n (a, 0) We have k = n —l - f ^ ± £ f + ^ ± ) ! / f (a t+ a ) a fc+1+ a a i+ a ^ ^ Then k=n k —n (a/c + a ) - n a — £ Gn (fl,a) ^ £ k= jfc=l (Theorem is proved) We denote £>n (a a ) = V “ ± Q)2 _|_ (Qn + tt)2 _ n a a fc + afc+ i + 2or an + a ! + 2a ’ where a ^ It follows that k = n —1 D n (a, 0) = 2 £ — p - + a“= G afc fc + G a fc-f fc+11 an + We will sharpen an inequality D n {a, 0) ầ ^ ' cifc — rt*^n(a ) fc=i where a* > 0, i = 1, n We obtain the following result N g u y e n Vu L u o n g 26 T h e o r e m 1.3 (i) D n(a, a ) is a non-increasing function (ii) D n (a, 0) ^ D n ( a , a ) ^ ^ S n (a ) Proof We have _ a + a fca + aị (afc + a ) Gfc + a j t + i + a a + afc + afc+ i o / (afc-Qfc+l 3afe — afc+ ^ a t J a + (Zfc + flfc+ It follows that n , n (a a ) n' ( n l fc^ , l ^ ^ afc “ \ V "' K ~ a * +i)2 , (fln -Q i)2 : a + a fc + dfc+ a + a n + ’ (fl/c afe+i)2 _ 1_ { a + afc + afc+i)2 (a n Qi ) < (2a + a n + a i ) — ~ Q Hence D n(a, a ) is a non-increasing monotonic function Since a ^ 0, D n (a ,a ) is a nonincreasing monotonic function Therefore D n (a, a) ^ D n (a, 0) D n (a, a ) ^ - S n (a) To complete the proof., we will show th a t We have n( \ = k\ ' i (Qfc + Q)2 [a i a ) 2_> afc+afc+ i + a k=l fc=n _ (a » + Q)2 an + a i + a na fc=n v -''/ \ nQ _ l ^ ( a k + a ) - — = ^ _ ak' k= fc=l Theorem is proved We denote \ = V ' _ (afc + a ) _ a’a " k= afc+ + p ũ k +2 + (1 + / ) a (an_ i + Q f ) a n + p a i + (1 + /ổ)a (an + a ) _ na a i + P a + (1 + ậ ) a + Ị3 (where variables Q ^ and (3 > is given ) It follows that Fn (a, 0) = áỉ ị-l an -— -1 -~ dk-ị-1 + Ị3dk+2 a n + /3ai We will sharpen the inequality Fn{ai 0) We obtain the following result ^ “ p S n (a) a i + (3a S h a rp en in g m e a n - f o r m a n d c y c lic in e q u a lity T heorem 1.4 (i) Fn (a:a ) is a non-increasing function (ii) Fn (a, 0) ^ F n ( a , a ) ^ j - ^ n (a) Proof We have (a k + o p _ Q2 + Ok a + ị GA + + 0a-k+ + (1 + ) a (1 + j3)a -f ajfc+ + Paic+ ( fifc fi + /2 + /? (1 + /3)2 Q _|_ _ ak + + /fafc + + /? + /3 (1 + /3)2 + P^k+ ) _ 2aic(ak+l -j-/?afc+ ) (1 + /5) a 4- a.k+ + P^k +2 / Qfe+l + /3Qfc+ _ \2 \ + /3_Q / (1 + p ) a + a fc+1 + pa k + ' Similary, we have / an an +/3a (1 + /?)2 + pa I _ \2 (an - i + op2 _ g a n 4- / ? « + (1 4- /3)a + /? a n -i + /3 + /? (1 + /3)a + an + /?ai k + 0-)^ _ a + /3a2 + (1 + p ) a l +p / Qj + /3q2 \2 2an _ Qị + /3a2 V + /? ~ fln/ + /3 (1 + /3)2 + (1 + /J)á + a x + /3a2 It follows that ( fln \^ ^ O] -f- /3o2 X? ( A_ 1^ , l l+ /J - “""V ^ n i a a ) = - > a* + - — - -f —-—tS L r (1 + /?)a + a n + /3d! (1 -f P)a + ữj + / a M + Pũk +2 „ , y>2 I + \2 ~ a *0 (1 + /?)a + flfc+ + (3ak+ Ị O'n P&2 /3gj (1 + P)F'Ji{a, a) = - J p + f ~ ~ n,,' ỵ _ _ Ì4 + ÌỈI!!Z [(1 + / ? ) a + a n + / ? O l ] / ak+i + {3ak + - V ^2 ((1 + / ) a + «1 * f / ? a 2]2 \ V + T a *) < [(1 + P)a + ak+l + (3ak+2}2 Hence, Fn ( a , a ) is a non-increasing monotonic function Since a ^ 0, we have F n (a, a ) < Fn (a,0) N g u y e n Vu L u on g 28 To complete the proof, we will show th a t Fn { a ,a ) > ; j- j - ^ S n (a) Indeed F„(a, a) > — tl V nQ ■Ẹ ( a t + a) - — Ảc=1 - 1 „ \ _ ^ a* Theorem is proved References p s Bullen D.S.Metrinovic’ and P.M Vasic’, Means and their inequalities, Reidel Publishing CO, Dordrecht - Boston 1988 G.H.Hardy, J.E Littewood, G.Polya, Inequalities , Cambridge University Press 1952 D.S.Metrinovic’ (with P.M Vasic’), Analytic inequalities, Springer Verlag, Berlin Heidelberg - New York 1970 G.V.Milovanovic’, Recent progress in Inequalities, Kluwer academic publishers 1996 ... a x + /3a2 It follows that ( fln ^ ^ O] -f- /3o2 X? ( A_ 1^ , l l+ /J - “""V ^ n i a a ) = - > a* + - — - -f - tS L r (1 + /?)a + a n + /3d! (1 -f P)a + ữj + / a M + Pũk... ) a + Ị3 (where variables Q ^ and (3 > is given ) It follows that Fn (a, 0) = áỉ ị-l an - -1 -~ dk- -1 + Ị3dk+2 a n + /3ai We will sharpen the inequality Fn{ai 0) We obtain... fc=i V è í 1/ > / *■ K - “ *+.) + (a- - l ) : a* + i+ ữ a i+ a We have G' (a, a ) = - ( a i >2 - £ (a + a i) " f a - at+ 1>2 ặ (a + ajt-fi)2 Hence G n (a, a) is a non-increasing monotonic function