We point out results connected with the log-convexity of power means of

two arguments

Sándor, József

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A note on log-convexity of power meansJózsef SándorBabeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romaniajsandor@math.ubbcluj.roSubmitted January 22, 2015 — Accepted September 29, 2015AbstractWe point out results connected with the log-convexity of power means oftwo arguments.1. IntroductionLet Mp(a, b) =(ap+bp2)1/p(p 6= 0), M0(a, b) =√ab denote the power mean (orHölder mean, see [2]) of two arguments a, b > 0. Recently A. Bege, J. Bukor andJ. T. Tóth [1] have given a proof of the fact that for a 6= b, the application p→Mpis log-convex for p ≤ 0 and log-concave for p ≥ 0. They also proved that it is alsoconvex for p ≤ 0. We note that this last result follows immediately from the well-known convexity theorem, which states that all log-convex functions are convex,too (see e.g. [2]). The proof of authors is based on an earlier paper by T. J. Mildorf(see [1]).In what follows, we will show that this result is well-known in the literature,even in a more general setting. A new proof will be offered, too.2. Notes and resultsIn 1948 H. Shniad [6] studied the more general means Mt(a, ξ) = (∑ni=1 ξiati)1/t(t 6= 0), M0(a, ξ) =∏ni=1 aξii , M−∞(a, ξ) = min{ai : i = 1, . . . }, M+∞(a, ξ) =max{ai : i = 1, . . . }; where 0 < ai < ai+1 (i = 1, . . . , n− 1) are given positive realnumbers, and ξi(i = 1, n) satisfy ξi > 0 and∑ni=1 ξi = 1.Put Λ(t) = logMt(a, ξ). Among other results, in [6] the following are proved:Annales Mathematicae et Informaticae45 (2015) pp. 107–110http://ami.ektf.hu107Theorem 2.1.(1) If ξ1 ≥ 12 then Λ(t) is convex for all t < 0.(2) If ξn ≥ 12 then Λ(t) is concave for all t > 0.Clearly, when n = 2, in case of Mp one has ξ1 = ξ2 = 12 , so the result by Bege,Bukor and Tóth [1] follows by Theorem 2.1.Another generalization of power mean of order two is offered by the Stolarskymeans (see [7]) for a, b > 0 and x, y ∈ R defineDx,y(a, b) =[y(ax−bx)x(ay−by)]1/(x−y), if xy(x− y) 6= 0,exp(− 1x + ax ln a−bx ln bax−bx), if x = y 6= 0,[ax−bx(ln a−ln b)]1/x, if x 6= 0, y = 0,√ab, if x = y = 0.The means Dx,y are called sometimes as the difference means, or extended means.Let Ix(a, b) = (I(ax, bx))1/x, where I(a, b) denotes the identic mean (see [2, 4])defined byI(a, b) = D1,1(a, b) =1e(bb/aa)1/(b−a) (a 6= b),I(a, a) = a.K. Stolarsky [7] proved also the following representation formula:logDx,y =1y − xy∫xlog Itdt for x 6= y.Now, in 2001 the author [4] proved for the first time that the application t→ log Itis convex for t < 0 and concave for t > 0.This in turn implies immediately (see also [3]) the following fact:Theorem 2.2.(1) If x > 0 and y > 0, then Dx,y is log-concave in both x and y.(2) If x < 0 and y < 0, then Dx,y is log-convex in both x and y.Now, remark thatMp(a, b) = D2p,p(a, b),so the log-convexity properties by H. Shniad are also particular cases of Theo-rem 2.2.We note that an application of log-convexity of Mp is given in [5].108 J. Sándor3. A new elementary proofWe may assume (by homogenity properties) that b = 1 and a > 1. Let f(p) =ln((ap + 1)/2)/p, and denote x = ap. Then, as x′ = dxdp = ap ln a = x ln a, from theidentity pf(p) = ln(x+ 1)/2 we get by differentiationf(p) + pf ′(p) =x ln ax+ 1. (3.1)By differentiating once again (3.1), we get2f ′(p) + pf ′′(p) =(x ln2 a)(x+ 1)− x2 ln2 a(x+ 1)2,which implies, by definition of f(p) and relation (3.1):p3f ′′(p) =(x ln2 x)(x+ 1)− x2 ln2 x(x+ 1)2− 2x+ 1[x lnx− (x+ 1) ln(x+ 12)]=x ln2 x+ 2(x+ 1)2 ln(x+ 1)− 2x(x+ 1) lnx(x+ 1)2,after some elementary computations, which we omit here. Putg(x) = x ln2 x+ 2(x+ 1)2 ln(x+ 12)− 2x(x+ 1) lnx.One has successively:g′(x) = ln2x+ 4(x+ 1) ln(x+ 12)− 4x lnx,g′′(x) =2 lnxx+ 4 ln(x+ 12)− 4 lnx,g′′′(x) = 2[1− lnxx2− 2x(x+ 1)]=−2x2(x+ 1)[x− 1 + (x+ 1) lnx].Now, remark that for x > 1, clearly g′′′(x) < 0, so g′′(x) is strictly decreasing,implying g′′(x) < g′′(1) = 0. Thus g′(x) < g′(1) = 0, giving g(x) < g(1) = 0.Finally, one gets f ′′(p) < 0, which shows that for x > 1 the function f(p) is strictlyconcave function of p. As x = ap with a > 1, this happend only when p > 0.For x < 1, remark that x − 1 < 0 and lnx < 0, so g′′′(x) > 0, and all aboveprocedure may be repeted. This shows that f(p) is a strictly convex function of pfor p < 0.References[1] A. Bege, J. Bukor, J. T. Tóth, On (log-)convexity of power mean, Annales Math.Inform., 42 (2013), 3–7.A note on log-convexity of power means 109[2] P. S. Bullen, Handbook of means and their inequalities, Kluwer Acad. Publ., 2003.[3] E. Neuman, J. Sándor, Inequalities involving Stolarsky and Gini means, Math.Pannonica, 14 (2003), no. 1, 29–44.[4] J. Sándor, Logaritmic convexity of t-modification of a mean, Octogon Math. Mag.,9 (2001) no. 2, 737–739.[5] J. Sándor, On the Leach-Sholander and Karamata theorems, Octogon Math. Mag.,11 (2003) no. 2, 542–544.[6] H. Shniad, On the convexity of mean value functions, Bull. Amer. Math. Soc., 54(1948), 770–776.[7] K. B. Stolarsky, The power and generalized logarithmic means, Amer. Math.Mouthly, 87 (1980), 545–548.110 J. Sándor

A note on log-convexity of power means

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A note on log-convexity of power means

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