Định Lý Gauss và ứng dụng

J Econ Inequal DOI 10.1007/s10888-009-9120-9 Gauss’ theorem concerning the center of gravity and its application Miodrag Tomi´c Received: 20 July 2009 / Accepted: 20 July 2009 © Springer Science + Business Media, LLC 2009 Keywords Gauss’ theorem · Center of gravity · Complex plane Section If aν are arbitrary points in the complex variable plane and αν are positive numbers (weights), then the complex number, represented by the expression: n ξn = ν=1 n αν a ν ν=1 , n = 2, 3, , αν which is the center of gravity of aν , as noted by Gauss,1 is situated either inside or on the boundary of the smallest convex polygon drawn around the points aν The exact geometric interpretation of this theorem is given later by H A Schwarz, and particularly by L Fejér [1] Gauss [2] See also Lucas [7] 2e S Translated from Serbian by Ivica Urban from volume 1, pages 31–40, of Vesnik Društva matematiˇcara i fiziˇcara Narodne Republike Srbije, with the kind permission of the editor of Matematiˇcki vesnik and of the Society of Mathematicians of Serbia M Tomi´c (B) University of Belgrade, Studentski Trg 1, 11000 Belgrade, Serbia e-mail: ivica@ijf.hr M Tomi´c Section Let there be n points aν in the complex plane (Fig 1), and n positive numbers αν Then, from: ξ2 = α1 a + α2 a α2 α1 = a1 + (a2 − a1 ) = a2 − (a2 − a1 ) , α1 + α2 α1 + α2 α1 + α2 it follows that ξ is situated on the line a1 a2 between points a1 and a2 , for all possible choices of numbers α > and α > 0, i.e that ξ lies inside the smallest convex polygon drawn around the points aν and a2 From: ξn = α1 a + α2 a + · · · + αn a n (α1 + α2 + · · · + αn−1 ) ξn−1 + αn an = α1 + α2 + · · · + αn (α1 + α2 + · · · + αn−1 ) + αn and from the preceding remark it follows that the point ξ n is situated on the line ξn−1 an−1 , i.e inside the smallest convex polygon around the aν , because the point ξ n−1 is also situated inside or on the boundary of that polygon Section From the latter, a well-known geometric construction of the center of gravity ξ n is obtained To obtain: ξ2 = α1 a + α2 a α1 + α2 we need to draw through the points a1 and a2 , in any direction (different from a1 a2 ), two parallel line segments in the mutually opposite directions towards a1 a2 , one Fig Gauss’ theorem concerning the center of gravity and its application beginning at a1 of length α , the other beginning at a2 of length α The intersection of the line going through these two new points with a1 a2 gives ξ As: ξn = An−1 ξn−1 + αn an An−1 + αn where An−1 = α1 + α2 + · · · + αn−1 , the preceding construction leads, after n steps, to the center of gravity ξ n Section Among many applications of this theorem, we mention Jensen’s well known theorem about convex functions Jensen2 has shown in 1905 that for every convex function ϕ(t) on an interval (a, b ) the following inequality is satisfied: ⎛ n ⎞ n αν xν α ϕ (xν ) ⎜ ν=1 ⎟ ν=1 ν ⎟≤ ϕ⎜ (A) n ⎝ n ⎠ αν αν ν=1 ν=1 where xν are numbers in the interval (a, b ) and the αν are any positive numbers Jensen’s proof of this inequality is completely analogous to Cauchy’s proof that the geometric mean is smaller than the arithmetic mean Cauchy’s first inequality: √ ab ≤ (a + b ) i.e n(a) + n(b ) a+b ≤ n 2 corresponds to Jensen’s inequality: ϕ x+y ≤ ϕ (x) + ϕ (y) that serves for defining functions which are convex from above In this way Jensen introduces the most general class of functions that satisfy inequality A Section As Jensen noted, this inequality generalizes some well-known and in analysis often used inequalities For example, for: ϕ (x) = x p , p > Jensen Loc [5] See also Pólya and Szegư [8] Cit 2: Pólya and Szegư, p 50 M Tomi´c we have: p n αν xν p−1 n ≤ ν=1 n αν αν xνp ν=1 ν=1 which is Buniakowsky’s inequality, usually called the Cauchy–Schwarz inequality ([4], p 16) For p = and αν2 xν = βν αν we obtain Cauchy’s inequality: n αν βν n n ≤ αν2 · ν=1 ν=1 βν2 ν=1 p Replacing xν in the inequality preceding the last one with βν αν / and raising both sides to the power 1/ p, one obtains: n ν=1 1−1/ p n αν1−1/ p ·βν1/ p ≤ αν · ν=1 1/ p n βν ν=1 which is Hölder’s inequality (and so on) Section Jensen’s inequality A follows directly from Gauss’s theorem about the center of gravity, reflecting the obvious fact that every polygon inscribed into a convex continuous curve is convex Let y = f (x) be any convex line and let a1 , a2 , , an be points lying on that line, with real parts (abscissae) x1 , x2 , , xn (Fig 2), and let a be the point on the curve whose real part (abscissa) is: n x∗ = ν=1 n αν xν ν=1 Fig αν Gauss’ theorem concerning the center of gravity and its application Since ak = xk + i f (xk ), it follows that: n ξn = ν=1 n n αν xν ν=1 +i ν=1 αν αν f (xν ) n ν=1 αν and: x∗ = Re {a} = Re {ξn } = α1 x1 + α x2 + · · · + α n xn α1 + α2 + · · · + αn Gauss’s theorem obviously implies: Im {a} ≤ Im {ξn } and this yields Jensen’s inequality: n f α x1 + α x2 + · · · + α n xn α1 + α2 + · · · + αn ≤ ν=1 αν f (xν ) n ν=1 αν Section M Petrovich [9] has provided an upper limit for the expression: n αν f (xν ) αν ν=1 From the above mentioned geometric interpretation we immediately obtain another upper bound that is achieved when all αν = 0, except α and α n It is obvious (from Fig 2) that this upper bound is the imaginary part (ordinate) of the point b , i.e that: n ν=1 n αν f (xν ) n ν=1 ≤ f (x1 ) + αν f (x1 ) − f (xn ) ν=2 n x − xn αν xν ν=1 αν Expanding the theorem of Petrovich, J Karamata has in the same issue of Publications offered one of the most general formulations of this theorem, that as a special case contains the following theorem: Theorem Let xν and Xν , ν = 1, 2, , n, be two sequences of numbers in an interval (a, b ) sorted in increasing order, i.e Xν ≤ Xν+1 and xν ≤ xν+1 M Tomi´c In order that the inequality: n n f (xν ) ≤ ν=1 f (Xν ) (1) ν=1 is satisfied for every convex function f (t), it is necessary and sufficient that: k k Xν ≤ ν=1 xν , for every k = 1, 2, , n − (2) ν=1 and n n Xν = xν ν=1 (3) ν=1 Several proofs4 of this theorem are analytical in nature; moreover, Karamata’s general theorem is stated in the form of a Stieltjes integral However, this proof can be derived also in an elementary way on the basis of the above mentioned geometric observation With this approach, the theorem can in fact be stated in a more precise way by replacing the equality sign in Eq with ≤ under the assumption that the function f (t) is convex and monotone Theorem Let xν and Xν , ν = 1, 2, , n, be the two sequences of numbers in an interval (a, b ) sorted in increasing order, i.e Xν ≤ Xν+1 and xν ≤ xν+1 and let f (t) be a convex function on (a, b ); then the inequality: n n f (xν ) ≤ ν=1 f (Xν ) (1) ν=1 holds: (a) if the function f (t) is decreasing and if : k k Xν ≤ ν=1 xν , for every k = 1, 2, , n (4) ν=1 or : (b) if the function f (t) is increasing and if : n n Xν ≥ ν=k xν , for every k = 1, 2, , n (5) ν=k Karamata [6], see also Hardy et al [3], L Fuchs, A new proof of an inequality of Hardy, Littlewood, Polya, and J Aczél, A generalization of a notion convex functions, and loc cit 1, p 83 Gauss’ theorem concerning the center of gravity and its application Although this theorem assumes more on the function f (t), i.e that it must be monotonic, it still in a certain sense contains Theorem 1, as we will show in Section 9, applying it separately to the decreasing and increasing parts of a convex function Section The proof of part (b) of Theorem is obtained directly from (a) by replacing an increasing function f (t) by the function f (b − t) and the sequences xν and Xν by the sequences b − xn+1−ν and b − Xn+1−ν , respectively Thus, it is sufficient to prove part (a) of Theorem By Eq 4, for k = 1, we have: X1 ≤ x1 , and from monotonicity of the function f (t) it follows that: f (X1 ) ≥ f (x1 ) Case If x2 ≤ X2 and if we denote by ( , ) and (ξ , ϕ ) the coordinates of the center of gravity T1 of the line segment M1 M2 , and the center of gravity t1 of the line Fig M Tomi´c segment m1 m2 respectively (see Fig 3), then from condition for k = 2, that can be written as: = x1 + x2 X1 + X2 ≤ = ξ1 2 it follows that: = f (x1 ) + f (x2 ) f (X1 ) + f (X2 ) ≥ = ϕ1 , 2 because in this case the line segment M1 M2 is above m1 m2 , and the abscissa of the center of gravity T1 is to the left of the abscissa ξ of the center of gravity t1 , and also the center of gravity T1 is above t1 Case If x2 ≥ X2 , then, because of monotonicity of the function f (t), we have that: f (X2 ) ≥ f (x2 ) , which, summing with: f (X1 ) ≥ f (x1 ) gives: f (X1 ) + f (X2 ) ≥ f (x1 ) + f (x2 ) Thus, in both cases this inequality is fulfilled Let us now apply the same argument to the points T1 , M3 , t1 and m3 Case If x3 ≤ X3 then the line segment t1 m3 lies under the line segment T1 M3 , and according to condition with k = we have: = x1 + x2 + x3 X1 + X2 + X3 ≤ = ξ2 3 Thus, we have: = f (x1 ) + f (x2 ) + f (x3 ) f (X1 ) + f (X2 ) + f (X3 ) ≥ = ϕ2 3 Case If x3 ≥ X3 , then because of monotonicity of the function f (t) we have f (X3 ) ≥ f (x3 ), which, adding to: f (X1 ) + f (X2 ) ≥ f (x1 ) + f (x2 ) gives: f (X1 ) + f (X2 ) + f (X3 ) ≥ f (x1 ) + f (x2 ) + f (x3 ) It is clear that this procedure can be extended to any number of points; hence, Theorem is proved Gauss’ theorem concerning the center of gravity and its application Section Let us show now that Theorem is a direct consequence of Theorem Assume therefore that the function f (t) is convex in an interval (a, b ), monotonically de¯ and monotonically increasing for x¯ < t < b It is clear that, creasing for a < t < x, because of convexity of the function f (t), there can exist only one point x¯ at which the curve changes the sense of monotonicity If to the left of the point x¯ there are p points Xν and q points xν , and if p = q, then the sequence Xν and the sequence xν can always be extended with r = | p − q| points ¯ such that the newly obtained sequences: x, Xν and xν , ν = 1, 2, , n + r have the same number of points in the interval (a, x), and therefore, also in the interval (x, b ) These two, new sequences, are: Case if p < q, r = q − p: ⎧ for ν = 1, 2, , p, ⎨ Xν for ν = p + 1, p + 2, , p + r = q, Xν = x¯ ⎩ Xν−r for ν = q + 1, q + 2, , n + r, and: ⎧ for ν = 1, 2, , q, ⎨ xν for ν = q + 1, q + 2, , q + r, xν = x¯ ⎩ xν−r for ν = q + r + 1, q + r + 2, , n + r; Case if p < q, r = p − q: ⎧ for ν = 1, 2, , p, ⎨ Xν for ν = p + 1, p + 2, , p = r, Xν = x¯ ⎩ Xν−r for ν = p + r + 1, p + r + 2, , n + r, and: ⎧ for ν = 1, 2, , q, ⎨ xν for ν = q + 1, q + 2, , q + r = p, xν = x¯ ⎩ xν−r for ν = p + 1, p + 2, , n + r Since by the assumption of Theorem 1, the sequences xν and Xν satisfy conditions and 3, it is easy to see: Case that: n+r n+r Xν = ν=1 xν , ν=1 because this equation reduces to Eq 3, as in it on the left and on the right hand side r added points x¯ cancel out; M Tomi´c ¯ satisfy Case that the points of the sequences Xν and xν lying in the interval (a, x) ¯ b ) satisfy condition condition 4, and those points which belong to the interval (x, In other words, if we denote by s the maximum of p and q, then: k k Xν ≤ ν=1 xν , for k = 1, 2, , s ν=1 and: n+r n+r Xν ≥ ν=k xν , for k = s + 1, s + 2, , n + r ν=k ¯ and On the other hand, the function f (t) is decreasing on the interval (a, x), ¯ b ); therefore according to Theorem 1, parts (a) and increasing on the interval (x, (b), we have: s s f xν ≤ ν=1 f Xν ν=1 and: n+r n+r f xν ≤ ν=s+1 f Xν ν=s+1 Summing these two inequalities we obtain: n+r n+r f xν ≤ ν=1 f Xν , ν=1 and this inequality reduces to inequality 1, because on its left and right sides r equal ¯ cancel out, which proves Theorem members f (x) References Fejér, L.: Über die Wurzel von kleinsten absoluten Betrage einer algebraischen Gleichung Math Annalen, Bd 65 (1908) Gauss, C.F.: Werke Bd Ed (1866) Hardy, G.H., Littlewood, J.F., Pólya, G.: Some simple inequalities satisfied by convex functions Mess Math LVIII, 149–152 (1929) Hardy, G.H., Littlewood, J.F., Pólya, G.: Inequalities Cambridge Univ Press (1934) Jensen, J.L.W.V.: Sur les fonctions convexes et les inégalités ente les valeurs moyennes Acta Math T 30, 175 (1905) Karamata, J.: Sur une inégalité relative aux fonctions convexes Publ Math T I (1932) Lucas, C.H.F.: C R de l’Acad De Paris (1866) Pólya, G., Szegư, G.: Aufgaben und Lehrsätze aus der Analysis I, pp 51–53 Berlin (1925) Petrovich, M.: Sur une fonctionnelle Publ Math Beograd, T L (1932)