Reformulation of Shapiro s inequality

Transcription

1 Iteratioal Mathematical Forum, Vol. 7, 2012, o. 43, Reformulatio of Shapiro s iequality Tafer Tariverdi Departmet of Mathematics, Harra Uiversity Saliurfa, Turkey ttariverdi@harra.edu.tr Abstract We reformulate Shapiro s iequality with elemetary mathematics ad preset some ew Shapiro type iequalities by givig examples with a aalytic proof. Mathematics Subject Classificatio: 26D05, 26D15, 26D20 Keywords: Cyclic iequality, Shapiro s iequality 1 Itroductio I 1954 H. S. Shapiro [1] cojectured that E(x) = x k x k+1 + x k+2 2 (P ()) (1) where x k 0, x k+1 + x k+2 > 0 ad x +k = x k for k N. Equality occurrig oly if all deomiators are equal. Studies o (1) have bee based o couterexamples ad aalytic proofs have give for small so far. It is cojectured that (1) is true for eve 12 ad false for eve 14 ad that it is true for odd 23 ad false for odd 25. We ow give a brief history of attempts o cojecture. Let λ() = 1 if x 1,x 2,...,x E(x). The λ() 1 2 (2) clearly. The case for =1, 2 is trivial. Several authors [4] proved that (2) is true for =3, 4, 5, 6. Diaada [3] proved that (2) is true for 6 differet from the previous oes. Mordell [4] cojectured that (1) is false for all 7, but later [5] proved that (1) is true for = 7. Nortover [2], ackowledged assitace from M. J. Lighthill, gave a couterexample for = 20.

2 2126 T. Tariverdi I [2, 4, 5, 6, 9, 27], it was proved that (1) is false for all eve 14 ad this result was also credited to Herschor ad Peck [25]. Zulauf [7, 8] proved that (1) is false for eve 14 ad is false for odd 53. Dojokovic [9] proved that P (8) is true. Raki [6] proved that the iequality (1) is false for large eough. Diaada [10] proved that (i) if P (m) is true, where m is eve, the p() is true for all m, ad (ii) if P (m) is false, where m is odd, the p() is false for all m. I the same paper a couterexample for P (27) was give ad thus (2) is false for all odd 27. Nowosad [11] aalytically proved that P (10) is true. Bushell ad Crave [12] also aalytically proved that P (10) is true ad thus is true for all 10 ad gave a couterexample for = 25. Goduova ad Levi [13] verified P (12) partly aalytically ad partly umerically. Recetly, Bushell ad Mcleod [14] proved aalytically that P (12) is true. Raki [6, 15] gave a lower boud for λ = λ() ad λ = λ() respectively. Prior to Raki s result, the oly lower boud was kow [8] for λ(24) = Diaada [16, 17] improved lower bouds, foud by Raki, to λ = λ() ad λ = λ() respectively. Zulauf [8, 18] showed that λ λ(24) < later [17] improved to λ λ(24) < ad also gave a couterexample for = 24. Basto [24] obtaied a lower boud which is a improvemet o Raki s origial result [15]. Drifeld [19] prove that λ = λ() = A aalytic result for the same boud, with some difficulties metioed, also occurred [20]. Malcolm [23] umerically gave a couterexample for = 25, Dayki [26] umerically showed that (1) is false for =14, 16, 25, 27, 40, 41, 50, 51, 60, 61, 110, 111 ad gave couterexamples for = 25, 111 ad also foud that λ λ(111) < Troesch [21, 22] umerically proved that P (13) ad P (23) are true. For more sophisticated aalysis ad a brief history o cojecture see [11, 14, 22, 27, 29]. 2 Mai Results Set F (k) = where k =1, 2,...,. The oe writes F (k) = x k x k+1 + x k+2 > 0, (3) x 1 x 2 x 3...x (x 2 + x 3 )...(x + x +1 )(x +1 + x +2 ).

3 Reformulatio of Shapiro s iequality 2127 Usig the arithmetic mea ad geometric mea iequality we reformulate Shapiro s iequality i terms of give data as x 1 x 2 x 3...x F (k) { (x 2 + x 3 )...(x + x +1 )(x +1 + x +2 ) }1/, (4) where x +1 = x 1 ad x +2 = x 2. Equality occurs if all x k s are equal. So we formally prove the followig theorem. Theorem 2.1. Let x k > 0 ad x +k = x k be for all k N. The x 1 x 2 x 3...x F (k) { (x 2 + x 3 )...(x + x +1 )(x +1 + x +2 ) }1/, (5) equality occurs if all x k s are equal. The followig result is immediately follows from the above theorem. Corollary 2.2. F (k) 1. 2 Proof. Applyig (x k + x k+1 ) 2 x k x k+1 (k =1, 2,...,) to the deomiator of (4), oe obtais (x 2 + x 3 )...(x + x +1 )(x +1 + x +2 ) 2 x 1 x 2 x 3...x 1 x. Therefore, F (k) 1 2. We will cosider x k > 0 i the followig lemmas where k =1, 2,...,. Lemma 2.3. E(x) is a homogeeous fuctio of degree 0. Proof. Proof is trivial. Lemma 2.4. E(x) satisfies differetial equatio x ke xk (x) =0 Proof. It is clear that E(x) possess cotiuous partial derivatives. x k E xk (x) =0 follows immediately from Lemma 2.3. For these type properties of E(x) see [11, 14, 27]. Lemma 2.5. Let (x k + x k+1 ) 2 max{x k,x k+1 }, x +1 = x 1 ad x +2 = x 2 be where k =1, 2,...,. The F (k) 1 2.

4 2128 T. Tariverdi Proof. If max{x k,x k+1 } = x k or x k+1 the (x k + x k+1 ) 2 x 1 x 2 x 3...x 1 x. Thus, F (k) 1 2. Theorem 2.6. If Lemma 2.5 holds the oe obtais Shapiro s iequality F (k) 2. Proof. Usig (4), proof follows immediately from Lemma Examples: Some ew Shapiro type iequalities We wat to look at the followig iterestig idetity [for proof, see [28, p.25]]. Example 3.1. Set 1 si( kπ )= 2 1 ( =2, 3,...). F (k) = si( kπ ) where k =1, 2,..., 1. The usig the above idetity together with (4) oe gets 1 1 F (k) ( 1)( si( kπ )) 1 1 sice 1 1/( 1) 2 as varies from 2 to. Example 3.2. Set =( 1)( 2 ) F (k) = si( kπ ) where k =1, 2,..., 1 ad x =1. The usig the above idetity together with (4) oe gets, 1 F (k) ( si( kπ )x ) 1 = ( 2 1 ) 1 = ( 2 2 ) 1 2, sice 1 (2) 1/ 2 as varies from 2 to.

5 Reformulatio of Shapiro s iequality 2129 Refereces [1] H. S. Shapiro, Advaced problem 4603, Amer. Math. Mothly 61(1954), pp [2] F. H Nortover, A ivalid iequality, Amer. Math. Mothly 63(1956), pp [3] P. H. Diaada, Extesios of a iequality of H. S. Shapiro, Amer. Math. Mothly 66(1959), pp [4] L. J. Mordell, O the iequality r=1 x r/(x r+1 + x r+2 ) ad some 2 others, Abh. Math. Sem. Uiv. Hamburgh 22(1958), pp [5] L. J. Mordell, Note o the iequality r=1 x r/(x r+1 + x r+2 ) 2, J. Lodo Math. Soc. 37(1962), pp [6] R. A. Raki, A iequality, Math. Gaz, 42(1958), pp [7] A. Zulauf, Note o cojecture of L. J. Mordell, Abh. Math. Sem. Uiv. Hamburgh 22(1958), pp [8] A. Zulauf, O a cojecture of L. J. Mordell II, Math. Gaz. 43(1959), pp [9] D. Z. Djokovic, Sur ue iegalite, Proc. Glasgow Math. Assoc. 6(1963), pp [10] P. H. Diaada, O a cyclic sum, Proc. Glasgow Math. Assoc. 6(1963), pp [11] P. Nowosad, Isoperimetric eigevalue problems i algebras, Comm. Pure Appl. Math. 21(1968), pp [12] P. J. Bushell ad A. H. Crave, O Shapiro s cyclic iequality, Proc. Roy. Soc. Ediburgh Sect. A 26(1975/76), pp [13] E. K. Goduova ad V. I. Levi, A cyclic sum with 12 terms, Math. Notes 19(1976), pp [14] P. J. Bushell ad J. B. Mcleod, Shapiro s cyclic iequality for eve, J. of Iequal. & Appl. 7(2002), pp [15] R. A. Raki, A cyclic iequality, Proc. Ediburgh Math. Soc. 12(1961), pp [16] P. H. Diaada, A cyclic iequality ad a extesio of it I, Proc. Ediburgh Math. Soc. 13(1962), pp

6 2130 T. Tariverdi [17] P. H. Diaada, A cyclic iequality ad a extesio of it II, Proc. Ediburgh Math. Soc. 13(1962), pp [18] A. Zulauf, Note o some iequalities, Math. Gaz. 43(1959), pp [19] V. G. Drifeld, A cyclic iequality, Math. Notes 9(1971), pp [20] B. A. Troesch, The shootig method applied to a cyclic iequality, Math. Comp. 34(1980), pp [21] B. A. Troesch, O Shapiro s cyclic iequality for = 13, Math. Comp. 45(1985), pp [22] B. A. Troesch, The validity of Shapiro s cyclic iequality, Math. Comp. 53(1989), pp [23] M. A. Malcolm, A ote o a cojecture of L. J. Mordell, Math. Comp. 25(1971), pp [24] V. J. Basto, Some cyclic iequalities, Proc. Ediburgh Math. Soc. 19(1974), pp [25] M. Herschor ad J. E. L. Peck, A ivalid iequality, Amer. Math. Mothly 67(1960), pp [26] D. E. Dayki, Iequalities of cyclic ature, J. Lodo Math. Soc. 3(1971), pp [27] P. J. Bushell, Shapiro s cyclic sum, Bull. Lodo Math. Soc. 26(1994), pp [28] M. R. Spiegel, Schaum s Outlie of Theory ad Problems of Complex Variables, McGraw-Hill, New York, [29] R. P. Lewis, Atisocial dier parties, Fiboacci Quart. 33(1995), pp Received: April, 2012