-0 and K any positive real number. Similarly we define,
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1 AN INEQUALITY OF SCHÜR AND AN INEQUALITY OF NEWTON K. V. MENON Let El H) denote the elementary complete symmetric functions of the rth order in ai, a2,, am respectively. If (1.1) et = e\ jl \ i / (m + r 1\ (1.2) *,«*,/( f ) then we have two well-known inequalities (1.3) El ^ Er-iEr+i (1.4) Hi Û Hr-lHr+l. The inequality (1.3) is known as Newton's inequality is true when cti, a2,, am are real numbers. The inequality (1.4), given by Professor Schur, is true when ai, a2, -, am are nonnegative reals. In [l], Whiteley has given the following generalizations of (1.3) (1.4): Let (1.5) Er(K) = ^A^X,-, \imoti<x'2-0 amm, ii + i im = r, K any positive real number. Similarly we define, (1.6) where Hr(K) = 2J àifii2 á,mai a2 am, ii + i2+ + im = r, >, = (-» (*) Received by the editors December 5,
2 442 K. V. MENON [August K any negative real number. In [l], it is proved that if i / (Km\ (1.7) r(a) = r(a) / (^ j, i /{Km\ (1.8) HriK)=HriK) /(^ j, then (1-9) [Er(K)]2^ET^(K)-Er+l(K), (1.10) [/7r(A)]2g/7r_!(A)-/7r+i(A). In this paper we prove that (1.9) (1.10) are true for a more general type of nonsymmetric functions (1.11) Er(Ki, A2,, Km) (1.12) Hr(Kh K2,, Km) which are defined in the following way: Let EliKi, K, Km) = (**) (**} - (^) aï1«;2 al", (1.13) ii + i2 + + im = r; i,, _ /Ai +! - 1\ /A2 + i2-1\ Hr(KuK2, -,*«) = ( H j (1-14) (Am /Am + îm Îm-1\ 1\,,-,,-, im xi a2 am, im / î'i + î2 + + m = f, where Ai, A2,, Km are positive real numbers. Now we define Tt(K K Ar(Ai, A2,, Km),r(,Ai, A2,, Am; - (1.15) /Ai + A2 + + K, HriKi, K2,, Km) HriKi, K2,, Km) (1.16) /A! /Ki + A2+ K2+ +Am+r-l\ fam+r-ln we note that
3 i969] INEQUALITIES OF SCHUR AND NEWTON 443 Er(K, K,K,,K) = Er(K) (1.17) V Hr(K, K,---,K)= HT(K), (1.18) Er(l, 1,, 1) = Er Hr(\, l,--,l) = HT. The generating series for (1.13) is (1.19) 1 + E ei(ki, Kz,--, Km)x = E (1 + ctixf ' -i the generating series for (1.14) is m (1.20) 1 + HT(Ki, Kt,--, Km)x = II (1 - «<*)" ' where Ku K2,, Km are positive real numbers. In the proof of the theorem we make use of the classical theory of maxima minima as in [l]. 2. Lemma 1. If r = \, then for all m, [Er(Ki, K2,, Km)]2 = HT-i(Ki, K2,, Km)Hr+i(Ki, K2,, Km) where alf a2,, am are nonnegative reals. The inequality is strict unless all the variables are equal. Proof. (2.2) Hi(Ki, Kt,--, Km) = (2.3) H2(Ki, K2,---, Km) = Hence (X/<=i Kioti) (Er-i Ki) H2(Ki, K2,, Km) - [Hi(Ki, K2,---, Km)]2 2i-i Ki(Ki + l)a,- +z2i^i 2KiKjaicti (X iffi)(i)s-itf.-+i) 1 ( sr_x Kn zr_, Ki + d (2.4) {( Kt) I" JTiCJCí + l)a! + 2 it.i^a/l i\ =i / L =i i^j J -(P'+1)(5S«)}-
4 444 K. V. MENON [August Or, 772(Ai, A2,, Km) [77i(A1; A2,, Km)\2 (2.5) 1 Hence the result. { Z KtK,(at - a,a. (ZT-iä-.OHZT-iA.+ i) V it*i I 3. Lemma 2. If r = l, then for all m, \Er(Ki, A2,, Am)]2 = ET-i(Ki, K2,, Km)ET+i(Ki, K2,, Am) where a.\, a2,, am are nonnegative reals. The inequality is strict unless all the variables are equal. Proof. Same as Lemma Lemma 3. For all r, if m = 1, then,.... Hr-l(Kl, K2,, Km)Hr+l(Kl, K2,, Km) (4.1) - [77r(Ai, A2,, Km)]2 = 0,.. Er-iiKi, K2,, Km)Er+iiKi, K2,, Am) (4-2) - [ r(ai, A2,, Am)]2 = 0. Proof. Follows from (1.15) (1.16). 5. Lemma 4. (5.1) A d i _, - HriKi, K2,, Km) i=i dai = ( Ê * i + r - l) 77r_i(Ai, A2,, Am), (5.2) A 3 i Z^ -EriKi, ;=1 3a,- K2,, Km) = ( í<-f+l) ET-i(Ku A2,, Km). Proof. By Euler's theorem on homogeneous functions we have (5.3) äi-hriki, K2,, Km) = r77r1(ai, A2,, Km).
5 1969] INEQUALITIES OF SCHUR AND NEWTON 445 From (1.20) (5.3) we get (5.1). In the same way (5.2) can be proved from (1.19). 6. Theorem 1. If «i, a2,, am are nonnegative reals, then (6.1) Hr-i(Ki, K2,, Km)Hr+i(Ki, K2,, Km) (6.2) Hr(Ki, Ki, > [Hr(Ki, K2,, Km)]2, Km)llr > HT+i(Ku K2,--, Kmyn'+» where r = l, 2, 3, are equal. The inequality is strict unless all the variables Proof. Let au a2,, am be m variables subject to the following conditions: ai > 0, ct2 > 0,, am > 0 (6.3) Hr-i(Ki, K2,, Km)Hr+i(Ki, K2,, Km) = 1. These conditions define a closed set of points in w-dimensional space. This set of points is bounded also since the function Br-l(Ki, K2, Km) flr+i(i^i, K2,, Km) has positive coefficients includes the term a2r for i= 1, 2,, m. Also from Lemma 1 the theorem is true for all m if r = l, by Lemma 3, the theorem is true for all r, if m i. We use double induction on m r to prove the theorem. We shall prove that it holds for all m r provided it holds for all pairs m, r with mi<m, for all pairs m, ri with ri<r. By Euler's theorem on homogeneous functions (6.3) (6.4) _ 01 1 ^ a i Hr-i(Ki, K2,, Km)Hr+i(Ki, K2, don 2rHr-i(Ki, K2, Km)Br+i(Ki, K2, Km), Km) = 2r. From (6.4) we note that the first partial derivatives cannot all vanish at the point at which [Hi (Ku K2,, Km)]2 takes its maximum value, say M. Applying Lagrange's conditions at this point we have
6 446 K. V. MENON [August (6.5) 3 r [77r(Ai, A2,, Am)J oai From (6.5) we have «i i X-Hr-i(Ki, K2,, Km)Hr+i(Ki, A2,, Km) 0. dcti i d i 2Hr(Ki, K2,, Km)-HT(Ki, d<xi K2,, Km) (6.6) = X \hliíku A2,, Km) h)+î(ki, K2,, Km) \ da, Now multiplying i d i \ + Hr+liKl, K2,, Km) -Hr-liKi, K2,, Km) >. doti ) each of (6.6) by a, adding we have i A d 1 2HriKi, K2,, Km) 2-i ai ~ HriKi, K2,, Km) <=i dan Í m ô X \Hr-liKl, K2,, Km) _/(Xi- Hr+liKl, K2,,Km) (6.7) ^ M d<xi But m d \ + Hr+i(Ki,K2,,Km) _, «<-Hr-i(Ki, K2,, Km)> = 0.,=i don ) A d i 2-, «< ~ ' HT+i(Ki, K2,, Km) (6.8) i da, Hence from (6.7) (6.8) we have = (r + l)hxr+i(ki, Ki, - -,Km). r i,2, «\ 2r[77r(Ai, A2,, Km)\ (6.9) \2rHT-iiKi, K2,, Km)Hr+iiKi, A2,, Km) 0 or, [HriKi, K2,, Km)\ (6.10) X [/íj_i(ai, A2,, Km)Hl+1iKi, K2,, Km)] Now using (6.6)
7 i969] INEQUALITIES OF SCHUR AND NEWTON 447 2h\(Ki, K2,---,Km)(f^Ki + r- lw^ä K2,, Km) (6.11) V" m ' = X jifti^i, Ki,, Km) ( Ki + r\ Hr(Ki, K2,, Km) + Hr(Ki, K2,, Km) ( Kt + r - 2 j H^2(Ki, K2,---, Km)}. Now from (6.11) we have, 2 { E Ki + r - l - X Ki + r\ (6.12) = X j Ki + r - 2Í Now from (6.12) (6.10) we have BT+i(Ki, K2,, Km)Hr-2(Ki, K2,, Km) HT(Ki, K2,, Km)Hr_1(Ki, K2,, Km) 2 { Ki + r - ll - X ÜT, + r\ (6.13) = i^ + r-2j But by induction hypothesis Hr(Ki, K2,, Km)Hr-2(Ki, K2,, Km) [HU(Ki, K2,, Km)\2 (6.14) IIr(Ki, K2,, Km)Hr-î(Ki, K2,, Km) [Hr_i(Ki, K2,---,Km)]2 (Er.i Ki + r- 1) (r - 1) > = (E"_, ü:.- + r - 2) r Hence
8 448 K. V. MENON [August 2( Kt + r - l\ - X( Kt + r) Therefore, or or a( ic, + >-2)LZÍ(^' + -'' 2 ( A< + f - 1 J - \( Af + rj ^ ÍEifilf-l) (ti r(+r-l~)^6x(' A', +,) (6.15, X<r + '(Z«*.+'-«r (I'íí + f) 1=1 In the next place we suppose that the maximum A7 is attained when «mi+1 = 0, ami+2 = 0,, am = 0, ai^o, -, ami? 0, (mi<m). Induction on m gives,,.,,. [Hr(Ki, Ki,, Kmi)] (6.16) X = - -; #.-i(ai, A2,, Kmi)Hr+l(Ki, K2,, Kmi) then clearly >al±i(g.r.»=i +,-i) r + 1 7=i A + r - 1 r + 1 7=1 A,- + r - 1 r 2m\Ki + r r? A,- + r ' t= 1 i= 1 hence the theorem. The second part of the theorem can be proved as in [2]. 7. Theorem 2. 7/«i, a2,, am are nonnegative reals, then [Er(Kl, Ki,---,Km)}2 > Er-i(Ki, Ki,, Km)Er+iiKi, K2,, Km)
9 i969] INEQUALITIES OF SCHUR AND NEWTON 449 Er(Ki, Ki,, Kmyi' > Er(Ki, K2,--, Km)l"+\ where r=l, 2,, r<i +l, where K is not integral K denotes the smallest of the real positive numbers, Ki, K2,, Km- The inequality is strict unless all the variables are equal. Proof. The restriction on r, makes all the terms positive, hence the theorem can be proved in the same way as Theorem 1. References 1. J. N. Whiteley, A generalization of a theorem of Newton, Proc. Amer. Math. Soc. 13 (1962), G. H. Hardy, J. E. Littlewood G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1952, p. 52. Dalhousie University
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