Novi Sad J. Math. Vol. 38, No. 3, 2008, 81-89 AN EXTENSION OF STOLARSKY MEANS Slavko Simi¶c1 Abstract. In this article we give an extension of the well known Sto- larsky means to the multi-variable case in a simple and applicable way. Some basic inequalities concerning this matter are also established with applications in Analysis and Probability Theory. AMS Mathematics Subject Classi¯cation (2000): 26A51, 60E15 Key words and phrases: Logarithmic convexity, extended mean values, generalized power means, shifted Stolarsky means 1. Introduction 1.1. There is a huge number of papers (cf. [2], [3], [6], [7], [8]) investigating properties of the so-called Stolarsky (or extended) two-parametric mean value, de¯ned for positive values of x; y by the following 8³ ´1=(s¡r) > r(xs¡ys) > s(xr ¡yr ) ; rs(r ¡ s) 6= 0 > ³ ´ > 1 xs log x¡ys log y <>exp ¡ s + xs¡ys ; r = s 6= 0 ³ ´ E (x; y) = s s 1=s r;s > x ¡y ; s 6= 0; r = 0 > s(log x¡log y) >p > xy; r = s = 0; :> x; x = y > 0: In this form it is introduced by Keneth Stolarsky in [1]. Most of the classical two variable means are special cases of the class E. x+y p For example, E1;2 = 2 is the arithmetic mean, E0;0 = xy is the geometric 1 x¡y x y x¡y mean, E0;1 = log x¡log y is the logarithmic mean, E1;1 = (x =y ) =e is the ³ ´1=r xr +yr identric mean, etc. More generally, the r-th power mean 2 is equal to Er;2r. Recently, several papers have been produced trying to de¯ne an extension of the class E to n; n > 2 variables. Unfortunately, this is done in a highly arti¯cial mode (cf. [4], [5], [9]), without a practical background. Here is an 1 Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia, e-mail: [email protected] 82 S. Simi¶c illustration of this point; recently J. Merikowski ([9]) has proposed the following generalization of the Stolarsky mean Er;s to several variables h s i 1 L(X ) s¡r E (X) := ; r 6= s; r;s L(Xr) where X = (x1; ¢ ¢ ¢ ; xn) is an n-tuple of positive numbers and Z Yn s sui L(X ) := (n ¡ 1)! xi du1 ¢ ¢ ¢ dun¡1: En¡1 i=1 The symbol En¡1 stands for the Euclidean simplex which is de¯ned by En¡1 := f(u1; ¢ ¢ ¢ ; un¡1): ui ¸ 0; 1 · i · n ¡ 1; u1 + ¢ ¢ ¢ + un¡1 · 1g: In this paper we give another attempt to generalize Stolarsky means to the multi-variable case in a simple and applicable way. The proposed task can be accomplished by founding a "weighted" variant of the class E, wherefrom the mentioned generalization follows naturally. In the sequel we shall need notions of the weighted geometric mean G = G(p; q; x; y) and weighted r-th power mean Sr = Sr(p; q; x; y), de¯ned by p q r r 1=r G := x y ; Sr := (px + qy ) ; where p; q; x; y 2 R+; p + q = 1; r 2 R=f0g: r r Note that (Sr) > (G) for x 6= y; r 6= 0 and limr!0 Sr = G. 1.2. We introduce here a class W of weighted two parameters means which includes the Stolarsky class E as a particular case. Namely, for p; q; x; y 2 R+; p + q = 1; rs(r ¡ s)(x ¡ y) 6= 0, we de¯ne ³ 2 s s ´ 1 ³ 2 s s ps qs ´ 1 r (Ss) ¡ (G) s¡r r px + qy ¡ x y s¡r W = Wr;s(p; q; x; y) := 2 r r = 2 r r pr qr : s (Sr) ¡ (G) s px + qy ¡ x y Various identities concerning the means W can be established; some of them are the following Wr;s(p; q; x; y) = Ws;r(p; q; x; y) ¡1 ¡1 Wr;s(p; q; x; y) = Wr;s(q; p; y; x); Wr;s(p; q; y; x) = xyWr;s(p; q; x ; y ); a a 1=a War;as(p; q; x; y) = (Wr;s(p; q; x ; y )) ; a 6= 0: Note that p ³r2 x2s + y2s ¡ 2( xy)2s ´1=2(s¡r) W (1=2; 1=2; x; y) = p 2r;2s s2 x2r + y2r ¡ 2( xy)2r An extension of Stolarsky means 83 ³r2 (xs ¡ ys)2 ´1=2(s¡r) = = E(r; s; x; y): s2 (xr ¡ yr)2 Hence E ½ W . The weighted means from the class W can be extended continuously to the domain D = f(r; s; x; y)jr; s 2 R; x; y 2 R+g: This extension is given by 8³ ´1=(s¡r) > r2 pxs+qys¡xpsyqs > s2 pxr +qyr ¡xpr yqr ; rs(r ¡ s)(x ¡ y) 6= 0 >³ ´ > s s ps qs 1=s > 2 px +qy ¡x y ; s(x ¡ y) 6= 0; r = 0 > pqs³ 2 log2(x=y) < s s exp ¡2 + px log x+qy log y Wr;s(p; q; x; y) = s pxs+qys¡xpsyqs´ > ps qs ; s(x ¡ y) 6= 0; r = s > ¡ (p log x+q log y)x y > pxs+qys¡xpsyqs > >x(p+1)=3y(q+1)=3; x 6= y; r = s = 0 :> x; x = y: 1.3. A natural generalization to the multi-variable case gives 8 ³ P Q p ´1=(s¡r) > r2( p xs¡( x i )s) > P i i Q i > 2 r pi r ; rs(s ¡ r) 6= 0; > s ( pixi ¡( xi ) ) >³ P Q p ´1=s > p xs¡( x i )s < 2 P i i P i 2 2 2 ; r = 0; s 6= 0; W (p; x) = s pi log xi¡( pi log xi) r;s ³ P s P Q pi s ´ > pix log xi¡( pi log xi)( x ) > ¡2 i P Q i >exp s + s pi s ; r = s 6= 0; > pixi ¡( xi ) > ³ P 3 P 3 ´ :> Ppi log xi¡( Ppi log xi) exp 2 2 ; r = s = 0: 3( pi log xi¡( pi log xi) ) n where x = (x1; x2; ¢ ¢ ¢ ; xn) 2 R+; n ¸ 2, p is an arbitrary positive weight sequence associated with x and Wr;s(p; x0) = a for x0 = (a; a; ¢ ¢ ¢ ; a). P Q Pn Qn We also write (¢); (¢) instead of 1 (¢); 1 (¢). The above formulae are obtained by an appropriate limit process, imply- ing continuity. For example Ws;s(p; x) = limr!s Wr;s(p; x) and W0;0(p; x) = lims!0 W0;s(p; x). 2. Results and applications Our main result is contained in the following Proposition 1. The means Wr;s(p; x) are monotone increasing in both vari- ables r and s. Passing to the continuous variable case, we get the following de¯nition of the class W¹ r;s(p; x). 84 S. Simi¶c Assuming that all integrals exist 8 ³ 2 R s R ´1=(s¡r) > r ( R p(t)x (t)dt¡exp(s R p(t) log x(t)dt)) > s2( p(t)xr (t)dt¡exp(r p(t) log x(t)dt) ; rs(s ¡ r) 6= 0; > R R >³ s ´1=s > 2 R p(t)x (t)dt¡exp(sR p(t) log x(t)dt) > s2 2 2 ; r = 0; s 6= 0; < ³p(t) log x(t)dt¡R( p(t) log x(t)dt) p(t)xs(t) log x(t)dt W¹ r;s(p; x) = ¡2 R R > exp s + p(t)xs(t)dt¡exp(s p(t) log x(t)dt) > R R ´ ; r = s 6= 0; > ( Rp(t) log x(t)dt) exp(Rs p(t) log x(t)dt) > ¡ s > R p(t)x (t)dt¡exp(R s p(t) log x(t)dt) > ³ 3 3 ´ :> Rp(t) log x(t)dt¡( Rp(t) log x(t)dt) exp 3( p(t) log2 x(t)dt¡( p(t) log x(t)dt)2) ; r = s = 0 whereR x(t) is a positive integrable function and p(t) is a non-negative function with p(t)dt = 1. >From our former considerations a very applicable assertion follows Proposition 2. W¹ r;s(p; x) is monotone increasing in either r or s. As an illustration we give the following Proposition 3. The function w(s) de¯ned by 8³ ´ < 1=s 12 (¡(1 + s) ¡ e¡γs) ; s 6= 0; w(s) := (¼s)2 : 4»(3) exp(¡γ ¡ ¼2 ); s = 0; is monotone increasing for s 2 (¡1; 1). In particular, for s 2 (¡1; 1) we have ¼s (¼s)4 ¡(1 ¡ s)e¡γs + ¡(1 + s)eγs ¡ · 1 ¡ ; sin(¼s) 144 where ¡(¢);»(¢); γ stands for the Gamma function, Zeta function and the Euler's constant, respectively. Applications in Probability Theory For a random variable X and an arbitrary distribution with support on (¡1; +1), it is well known that EeX ¸ eEX : k Denoting the central moment of order k by ¹k = ¹k(X) := E(X ¡ EX) , we improve the above inequality to the following Proposition 4. For an arbitrary probability law with support on R, we have X EX Ee ¸ (1 + (¹2=2) exp (¹3=3¹2))e : An extension of Stolarsky means 85 Proposition 5. We also have that ³EesX ¡ esEX ´1=s 2 2 s σX =2 is monotone increasing in s. Especially interesting is studying of the shifted Stolarsky means E¤, de¯ned by ¤ Er;s(x; y) := lim Wr;s(p; q; x; y): p!0+ Their analytic continuation to the whole (r; s) plane is given by 8 ³ 2 s s ´1=(s¡r) > r (x ¡y (1+s log(x=y))) ; rs(r ¡ s)(x ¡ y) 6= 0; > s2(xr ¡yr (1+r log(x=y))) >³ ´1=s > 2 xs¡ys(1+s log(x=y)) < s2 log2(x=y) ; s(x ¡ y) 6= 0; r = 0; ¤ ³ ´ E (x; y) = s s s r;s >exp ¡2 + (x ¡y ) log x¡sy log y log(x=y) ; s(x ¡ y) 6= 0; r = s; > s xs¡ys(1+s log(x=y)) > >x1=3y2=3; r = s = 0; :> x; x = y: Main results concerning the means E¤ are the following ¤ Proposition 6.