JOURNAL OF SCIENCE AND ARTS ă GENERALIZATIONS AND REFINEMENTS FOR BERGSTROM AND RADON’S INEQUALITIES ˘ DORIN MARGHIDANU Abstract In the present work there are pointed and demonstrated some generalizations and refinements for Bergstră om and Radons inequalities But not before making some historical remaks on the parenthood of these inequalities We present a new demonstration and a refinement for Radon’s inequality, which is based on a recently initiated method, using the monotony of a sequence associated to the inequality Some applications are also presented Keywords: Bergstră om inequality, C-B-S inequality, Radon inequality, power-means inequality, refinement Mathematics Subject Classification : 26D15 It is well-known and very often used lately- Bergstră oms inequality ( see [7] , [11] , [14] ) , namely , Proposition (Bergstră oms inequality) If x k ∈ R , ak > 0, k ∈ {1, 2, , n} , then the following inequality holds , x21 x2 x2 (x1 + x2 + xn )2 + + + n ≥ , a1 a2 an a1 + a2 + an (1) with equality for : xa11 = xa22 = = xann It is equivalent with Cauchy-Buniakowski-Schwarz inequality For the less simple implication, Bergstră om inequality ⇒ C-B-S inequality, see [2], [5], [20] This inequality is often called Titu Andreescu’s inequality (or Andreescu lemma –presented in [1] , having as base a problem published by the first author in the RMT journal, in 1979 ), or Engel’s inequality (or Cauchy-Schwarz inequality in Engel form - in Germanofon mathematical literature , [12] ) In fact , this inequality , for the case n = was enounced by H Bergstrăom in 1949 , in the more general frame of complex number modules , from denominators and in more relaxed conditions , for nominators (see [7] , [14 ] , [11] ) : • Let z1 , z2 ∈ C and u, v ∈ R such that u = 0, v = 0, u + v = Then we have: (2) i) |z1 + z2 |2 |z1 |2 |z2 |2 + ≥ u v u+v , if 1 + >0 u v ; |z1 |2 |z2 |2 |z1 + z2 |2 1 (3) ii) + ≤ , if + < u v u+v u v The equality holds if and only if zu1 = zv2 More than that , the inequality (1) , is a particular case of some of Radon’s inequality, discovered ever since 1913 , ( see [19] , [9] and rediscovered (? ) in [16] and [6] ) Proposition (Radon’s inequality) If ak , xk > 0, p > 0, k ∈ {1, 2, , n}, then the following inequality holds, 57 JOURNAL OF SCIENCE AND ARTS p+1 n n (4) k=1 xk xp+1 k ≥ apk k=1 n , p ak k=1 with equality for : xa11 = xa22 = = xann Clearly , p = - Bergstrăoms inequality is obtained There are known some demonstrations of Radon’s inequality, by using Hă olders inequality ([9], [16] ) , or by using the mathematical induction, [6] In what is to follow , we are going to demonstrate Radon’s inequality through a method recently initiated in [13] , which uses the monotony of an associated sequence Proof p+1 p+1 xp+1 xp+1 + +xn ) Let the sequence , dn := a1 p + a2 p + + xanp − (x(a1 +x +a + +a )p , n n for which we are going to prove that dn ≥ , for any n ≥ For this we are going to demonstrate something more, namely that (dn )n is an increasing monotonous sequence Indeed , we have , p+1 n+1 n+1 dn+1 − dn = k=1 xk xp+1 k − apk n k=1 n+1 p k=1 ak xk xp+1 k + apk − k=1 n p+1 ak p n+1 xk k=1 n = + ak p k=1 k=1 n p+1 n xk xn+1 n+1 − apn+1 k=1 n+1 ≥0 ak k=1 k=1 For the last inequality , Radon’s inequality has been used , for n = , (α + β)p+1 αp+1 β p+1 + ≥ , ap bp (a + b)p (5) n with : α = n xk , β = xn+1 , a= k=1 ak , b = an+1 k=1 (For the demonstration of the inequality (5) , see [6] ) It results that , dn ≥ dn−1 ≥ ≥ d2 ≥ d1 = (6) Application If a, b, c ∈ R+ , then , (7) √ a2 a b c +√ +√ ≥1 2 + 8bc b + 8ca c + 8ab ( The 42nd OIM, Washington D.C., 2001, Problem ) We write the left member of the inequality under the form , 3 a2 b2 c2 Ms := √ +√ +√ 3 a + 8abc b + 8abc c + 8abc 58 = JOURNAL OF SCIENCE AND ARTS and Radon’s inequality is applied for n = 3, xp+1 xp+1 xp+1 (x1 + x2 + x3 )p+1 + + ≥ , (a1 + a2 + a3 )p ap1 ap2 ap3 with the substitutions: x1 → a , x2 → b , x3 → c; a1 → a3 +8abc , a2 → b3 +8abc , a3 → c3 +8abc and p =1/2 It is obtained, Ms ≥ (a + b + c) (a3 + b3 + c3 + 24abc) = (a + b + c)3 ≥ a3 + b3 + c3 + 24abc The last inequality is reduced – after some simple calculations , to the obvious inequality , a(b2 + c2 ) + a(b2 + c2 ) + a(b2 + c2 ) ≥ 6abc The demonstration method given previously and in [13], also underlines an interesting method of refining the inequalities, which we can also be seen in the following theorem, Theorem (for refinement of Radon’s inequality) For ak , xk > 0, p ≥ 1, k ∈ {1, 2, , n}, n ∈ N≥2 , the inequality takes place, p+1 n n (8) k=1 xp+1 k ≥ apk xk k=1 n p ak + max 1≤i 0, p > 0, q ≥ 1, k ∈ {1, 2, , n}, then the inequality takes place, p+q n n (13) k=1 xk xp+q k ≥ q−1 · n apk k=1 n p , ak k=1 x1 a1 x2 a2 xn an with equality for: = = = Proof Using Radon’s inequality and the previous lemma, we successively have: n k=1 xp+q k = apk p+1 p+q p+1 xk n (4) ≥ apk k=1 n p+q k=1 p+1 xkp+1 (11) ≥ p n ak k=1 (11) ≥ p+q −1 n p+1 p+q p+1 n · p+1 k=1 = p n ak k=1 p+q n xk nq−1 xk · k=1 n p ak k=1 For q = in Theorem 8, Radon’s inequality is obtained, and for p = q = 1, Bergstră oms inequality is obtained 60 JOURNAL OF SCIENCE AND ARTS Corollary (the generalization of Radon’s inequality - a variant) If ak , xk > 0, , k ∈ {1, 2, , n}, p > 0, r ≥ p + 1, then the inequality holds, r n n (14) k=1 xrk p ≥ r−p−1 · n ak xk k=1 n p, ak k=1 with equality for : xa11 = xa22 = = xann Proof Noting r := p + q in Theorem 8, r ≥ p+1 results , and q − = r − p − 1, hence the enounce A similar result to the one in relation (14) is obtained in [17], using Jensen’s inequality 10 Application If a, b, c are the sides of a triangle and r ≥ , then , br cr (a + b + c)r−1 ar + + ≥ b+c−a c+a−b a+b−c 3r−2 or with the triangle known notations, we have , (15) (16) ar br cr 2r−1 + + ≥ r−2 · pr−1 p−a p−b p−c , Using the above inequality extensions, numberless other inequalities , such as those in : [1] , [2] , [3] , [16] , [17] – can be proved or generalized New ones can also be obtained References [1] Andreescu T., Enescu B., Mathematical Olympiad Treasures, Birkhauser, 2003 [2] Andreescu T., Lascu M., Asupra unei inegalit˘ a¸ti, Gazeta Matematic˘ a , seria B, Anul CVI, nr 9-10, pp 322-326, 2001 [3] Becheanu M., Enescu B., Inegalit˘ a¸ti elementare ¸si mai put¸in elementare, Editura Gil, Zalau, 2002 [4] Beckenbach E.F & Bellman R., Inequalities, Springer–Verlag, Berlin-Heidelberg-New York, 1961 [5] Bencze M., A New Proof of the Cauchy-Bunjakovski-Schwarz Inequality, OCTOGON Mathematical Magazine, Vol 10, No 2, pp.841- 842, October, 2002 [6] Bencze M., Inequalities Connected to the Cauchy-Schwarz Inequality, OCTOGON Mathematical Magazine, Vol 15, No 1, pp.58- 62, April, 2007 [7] Bergstră om H., A triangle - inequality for matrices, in: Den Elfte Skandinaviske Matematikerkongress, CityTrondheim,1949, Johan Grundt Tanums Forlag , pp.115-118, CityplaceOslo,1952 [8] Bullen P S.& Mitrinovi D S & Vasi P M., Means and Their Inequalities, D Reidel Publidshing Company, Dordrecht/Boston, 1988 [9] Bullen P S., Handbook of Means and Their Inequalities, Kluwer Academic Publishers, Dordrecht/Boston/London, 2003 [10] Dragomir S S., A Survey on Cauchy-Buniakowsky-Schwartz Type Discrete Inequalities, january 10, Melbourne, 2003 [11] Florea A., Niculescu P.C., Asupra inegalit˘ a¸tilor lui Bergstră om, Gazeta Matematic a , seria B, Anul CVII , nr 11, 2002 [12] Grinberg D., Arthur Engel’s minima principle and the Cauchy-Schwarz inequality / Andreescu Lemma , http://www.artofproblemsolving.com/Forum/viewtopic.php?p=107331#107331 [13] M˘ arghidanu D., Daz-Barrero J.L., R˘ adulescu S., New Refinements of Some Classical Inequalities, (send for publication) [14] Mitrinovi D S (in cooperation with Vasi P M ), Analytic Inequalities, Springer–Verlag, Band 165, StateplaceBerlin , 1970 [15] Mitrinovi D S., Pecaric J.E., Fink A.M., Classical and New Inequalities in Analysis, Kluwer Acad Press., 1993 61 JOURNAL OF SCIENCE AND ARTS [16] Panaitopol L., Consecint¸e ale inegalit˘ a¸tii lui Holder, Gazeta Matematic˘ a , seria B, Anul CVII , nr 4, pp 145-147, 2002 [17] Papacu N., Generaliz˘ ari ale unor inegalit˘ a¸ti, Arhimede , nr 5-6, pp 2-8, 2003 [18] R˘ adulescu S., Daz-Barrero J.L., Problema 9, cls a-X-a, Concursul Nat¸ional ARHIMEDE, etapa final˘ a, Bucure¸sti, 12 mai, 2007 [19] Radon J., ber die absolut additiven Mengenfunktionen, Wiener − Sitzungsber., (IIa), vol 122, p 1295-1438, 1913 [20] Steele J M., Cauchy-Schwarz Inequality: Yet Another Proof Colegiul Nat ¸ ional “A.I Cuza”, Corabia E-mail address: d.marghidanu@gmail.com 62 ... References [1] Andreescu T., Enescu B., Mathematical Olympiad Treasures, Birkhauser, 2003 [2] Andreescu T., Lascu M., Asupra unei inegalit˘ a ti, Gazeta Matematic˘ a , seria B, Anul CVI, nr 9 -10, pp... Pecaric J.E., Fink A. M., Classical and New Inequalities in Analysis, Kluwer Acad Press., 1993 61 JOURNAL OF SCIENCE AND ARTS [16] Panaitopol L., Consecint¸e ale inegalit˘ a tii lui Holder, Gazeta... 3 a2 b2 c2 Ms := √ +√ +√ 3 a + 8abc b + 8abc c + 8abc 58 = JOURNAL OF SCIENCE AND ARTS and Radon’s inequality is applied for n = 3, xp+1 xp+1 xp+1 (x1 + x2 + x3 )p+1 + + ≥ , (a1 + a2 + a3 )p ap1