International Mathematical Forum, Vol. 7, 2012, no. 43, 2125 - 2130
Reformulation of Shapiro’s inequality Tanfer Tanriverdi Department of Mathematics, Harran University Sanliurfa, 63300 Turkey
[email protected] Abstract We reformulate Shapiro’s inequality with elementary mathematics and present some new Shapiro type inequalities by giving examples with an analytic proof.
Mathematics Subject Classification: 26D05, 26D15, 26D20 Keywords: Cyclic inequality, Shapiro’s inequality
1
Introduction
In 1954 H. S. Shapiro [1] conjectured that E(x) =
n k=1
xk n ≥ xk+1 + xk+2 2
(P (n))
(1)
where xk ≥ 0, xk+1 + xk+2 > 0 and xn+k = xk for k ∈ N. Equality occurring only if all denominators are equal. Studies on (1) have been based on counterexamples and analytic proofs have given for small n so far. It is conjectured that (1) is true for even n ≤ 12 and false for even n ≥ 14 and that it is true for odd n ≤ 23 and false for odd n ≥ 25. We now give a brief history of attempts on conjecture. Let λ(n) =
1 1 inf x1 ,x2 ,...,xn E(x). Then λ(n) ≤ n 2
(2)
clearly. The case for n = 1, 2 is trivial. Several authors [4] proved that (2) is true for n = 3, 4, 5, 6. Diananda [3] proved that (2) is true for n ≤ 6 different from the previous ones. Mordell [4] conjectured that (1) is false for all n ≥ 7, but later [5] proved that (1) is true for n = 7. Nortover [2], acknowledged assitance from M. J. Lighthill, gave a counterexample for n = 20.
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In [2, 4, 5, 6, 9, 27], it was proved that (1) is false for all even n ≥ 14 and this result was also credited to Herschorn and Peck [25]. Zulauf [7, 8] proved that (1) is false for even n ≥ 14 and is false for odd n ≥ 53. Dojokovic [9] proved that P (8) is true. Rankin [6] proved that the inequality (1) is false for large enough n. Diananda [10] proved that (i) if P (m) is true, where m is even, then p(n) is true for all n ≤ m, and (ii) if P (m) is false, where m is odd, then p(n) is false for all n ≥ m. In the same paper a counterexample for P (27) was given and thus (2) is false for all odd n ≥ 27. Nowosad [11] analytically proved that P (10) is true. Bushell and Craven [12] also analytically proved that P (10) is true and thus is true for all n ≤ 10 and gave a counterexample for n = 25. Godunova and Levin [13] verified P (12) partly analytically and partly numerically. Recently, Bushell and Mcleod [14] proved analytically that P (12) is true. Rankin [6, 15] gave a lower bound for λ = λ(n)n→∞ ≥ 0.3047 and λ = λ(n)n→∞ ≥ 0.330232 respectively. Prior to Rankin’s result, the only lower bound was known [8] for λ(24) = 0.49950317. Diananda [16, 17] improved lower bounds, found by Rankin, to λ = λ(n)n→∞ ≥ 0.457107 and λ = λ(n)n→∞ ≥ 0.461238 respectively. Zulauf [8, 18] showed that λ ≤ λ(24) < 0.49950317 later [17] improved to λ ≤ λ(24) < 0.499197 and also gave a counterexample for n = 24. Baston [24] obtained a lower bound which is an improvement on Rankin’s original result [15]. Drinfeld [19] prove that λ = λ(n)n→∞ = 0.4945668. An analytic result for the same bound, with some difficulties mentioned, also occurred [20]. Malcolm [23] numerically gave a counterexample for n = 25, Daykin [26] numerically showed that (1) is false for n = 14, 16, 25, 27, 40, 41, 50, 51, 60, 61, 110, 111 and gave counterexamples for n = 25, 111 and also found that λ ≤ λ(111) < 0.49656. Troesch [21, 22] numerically proved that P (13) and P (23) are true. For more sophisticated analysis and a brief history on conjecture see [11, 14, 22, 27, 29].
2
Main Results
Set F (k) =
xk > 0, xk+1 + xk+2
where k = 1, 2, . . . , n. Then one writes n k=1
F (k) =
x1 x2 x3 . . . xn . (x2 + x3 ) . . . (xn + xn+1 )(xn+1 + xn+2 )
(3)
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Reformulation of Shapiro’s inequality
Using the arithmetic mean and geometric mean inequality we reformulate Shapiro’s inequality in terms of given data as n
F (k) ≥ n{
k=1
x1 x2 x3 . . . xn }1/n , (x2 + x3 ) . . . (xn + xn+1 )(xn+1 + xn+2 )
(4)
where xn+1 = x1 and xn+2 = x2 . Equality occurs if all xk ’s are equal. So we formally prove the following theorem. Theorem 2.1. Let xk > 0 and xn+k = xk be for all k ∈ N. Then n
F (k) ≥ n{
k=1
x1 x2 x3 . . . xn }1/n , (x2 + x3 ) . . . (xn + xn+1 )(xn+1 + xn+2 )
(5)
equality occurs if all xk ’s are equal. The following result is immediately follows from the above theorem. Corollary 2.2. nk=1 F (k) ≤ 21n . √ Proof. Applying (xk + xk+1 ) ≥ 2 xk xk+1 (k = 1, 2, . . . , n) to the denominator of (4), one obtains (x2 + x3 ) . . . (xn + xn+1 )(xn+1 + xn+2 ) ≥ 2n x1 x2 x3 . . . xn−1 xn . Therefore, nk=1 F (k) ≤ 21n . We will consider xk > 0 in the following lemmas where k = 1, 2, . . . , n. Lemma 2.3. E(x) is a homogeneous function of degree 0. Proof. Proof is trivial. Lemma 2.4. E(x) satisfies differential equation
n
k=1
xk Exk (x) = 0
Proof. It is clear that E(x) possess continuous partial derivatives. n
xk Exk (x) = 0
k=1
follows immediately from Lemma 2.3. For these type properties of E(x) see [11, 14, 27]. Lemma 2.5. Let (xk + xk+1 ) ≤ 2 max{xk , xk+1 }, xn+1 = x1 and xn+2 = x2 be where k = 1, 2, . . . , n. Then n k=1
F (k) ≥
1 . 2n
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T. Tanriverdi
Proof. If max{xk , xk+1 } = xk or xk+1 then n
n
(xk + xk+1 ) ≤ 2 x1 x2 x3 . . . xn−1 xn . Thus,
k=1
n
F (k) ≥
k=1
1 . 2n
Theorem 2.6. If Lemma 2.5 holds then one obtains Shapiro’s inequality n
F (k) ≥
k=1
n . 2
Proof. Using (4), proof follows immediately from Lemma 2.5.
3
Examples: Some new Shapiro type inequalities
We want to look at the following interesting identity [for proof, see [28, p.25]]. n−1
sin(
k=1
kπ n ) = n−1 n 2
(n = 2, 3, . . . ).
Example 3.1. Set F (k) = sin(
kπ ) n
where k = 1, 2, . . . , n − 1.
Then using the above identity together with (4) one gets n−1
F (k) ≥ (n − 1)(
k=1
n−1
sin(
k=1
1 kπ 1 n n−1 )) n−1 = (n − 1)( n−1 ) n−1 ≥ , n 2 2
since 1 ≤ n1/(n−1) ≤ 2 as n varies from 2 to ∞. Example 3.2. Set F (k) = sin(
kπ ) n
where k = 1, 2, . . . , n − 1 and xn = 1.
Then using the above identity together with (4) one gets n k=1
n−1
F (k) ≥ n(
k=1
sin(
1 kπ n 1 2n 1 n )xn ) n = n( n−1 ) n = n( n ) n ≥ , n 2 2 2
since 1 ≤ (2n)1/n ≤ 2 as n varies from 2 to ∞.
Reformulation of Shapiro’s inequality
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References [1] H. S. Shapiro, Advanced problem 4603, Amer. Math. Monthly 61(1954), pp. 571. [2] F. H Nortover, An invalid inequality, Amer. Math. Monthly 63(1956), pp.191-92. [3] P. H. Diananda, Extensions of an inequality of H. S. Shapiro, Amer. Math. Monthly 66(1959), pp. 489-491. n n [4] L. J. Mordell, On the inequality r=1 xr /(xr+1 + xr+2 ) ≥ 2 and some others, Abh. Math. Sem. Univ. Hamburgh 22(1958), pp. 229-240. n n [5] L. J. Mordell, Note on the inequality r=1 xr /(xr+1 + xr+2 ) ≥ 2 , J. London Math. Soc. 37(1962), pp. 176-178. [6] R. A. Rankin, An inequality, Math. Gaz, 42(1958), pp. 39-40. [7] A. Zulauf, Note on conjecture of L. J. Mordell, Abh. Math. Sem. Univ. Hamburgh 22(1958), pp. 240-241. [8] A. Zulauf, On a conjecture of L. J. Mordell II, Math. Gaz. 43(1959), pp. 182-184. [9] D. Z. Djokovic, Sur une inegalite, Proc. Glasgow Math. Assoc. 6(1963), pp. 1-10. [10] P. H. Diananda, On a cyclic sum, Proc. Glasgow Math. Assoc. 6(1963), pp. 11-13. [11] P. Nowosad, Isoperimetric eigenvalue problems in algebras, Comm. Pure Appl. Math. 21(1968), pp. 401-465. [12] P. J. Bushell and A. H. Craven, On Shapiro’s cyclic inequality, Proc. Roy. Soc. Edinburgh Sect. A 26(1975/76), pp. 333-338. [13] E. K. Godunova and V. I. Levin, A cyclic sum with 12 terms, Math. Notes 19(1976), pp. 510-517. [14] P. J. Bushell and J. B. Mcleod, Shapiro’s cyclic inequality for even n, J. of Inequal. & Appl. 7(2002), pp. 331-348. [15] R. A. Rankin, A cyclic inequality, Proc. Edinburgh Math. Soc. 12(1961), pp. 139-147. [16] P. H. Diananda, A cyclic inequality and an extension of it I, Proc. Edinburgh Math. Soc. 13(1962), pp. 79-84.
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[17] P. H. Diananda, A cyclic inequality and an extension of it II, Proc. Edinburgh Math. Soc. 13(1962), pp. 143-152. [18] A. Zulauf, Note on some inequalities, Math. Gaz. 43(1959), pp. 42-44. [19] V. G. Drinfeld, A cyclic inequality, Math. Notes 9(1971), pp. 68-71. [20] B. A. Troesch, The shooting method applied to a cyclic inequality, Math. Comp. 34(1980), pp. 175-184. [21] B. A. Troesch, On Shapiro’s cyclic inequality for n = 13, Math. Comp. 45(1985), pp. 199-207. [22] B. A. Troesch, The validity of Shapiro’s cyclic inequality, Math. Comp. 53(1989), pp. 657-664. [23] M. A. Malcolm, A note on a conjecture of L. J. Mordell, Math. Comp. 25(1971), pp. 375-377. [24] V. J. Baston, Some cyclic inequalities, Proc. Edinburgh Math. Soc. 19(1974), pp. 115-118. [25] M. Herschorn and J. E. L. Peck, An invalid inequality, Amer. Math. Monthly 67(1960), pp. 87-88. [26] D. E. Daykin, Inequalities of cyclic nature, J. London Math. Soc. 3(1971), pp. 453-462. [27] P. J. Bushell, Shapiro’s cyclic sum, Bull. London Math. Soc. 26(1994), pp. 564-574. [28] M. R. Spiegel, Schaum’s Outline of Theory and Problems of Complex Variables, McGraw-Hill, New York, 1999. [29] R. P. Lewis, Antisocial dinner parties, Fibonacci Quart. 33(1995), pp. 368-370. Received: April, 2012
