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... The logarithmic mean (L mean ) of θ, φ ∈ R + , which is of interest in many fields such as engineering, statistics, geometry, and thermodynamics (for more details, see [1][2][3]), is defined in the following integral formula ...
... Following [1], the L mean θ, φ is continuous when θ = φ and homogeneous and symmetric in θ and φ. Further, for θ, φ ∈ R + , we have ...
... Recently, various studies and generalizations for the L mean have been presented by several researchers (see e.g., [1][2][3][4][5][6]). ...
This paper aimed to obtain generalizations of both the logarithmic mean (L mean) and the Euler's beta function (EBF), which we call the extended logarithmic mean (EL mean) and the extended Euler's beta-logarithmic function (EEBLF), respectively. Also, we discussed various properties, including functional relations, inequalities, infinite sums, finite sums, integral formulas, and partial derivative representations, along with the Mellin transform for the EEBLF. Furthermore, we gave numerical comparisons between these generalizations and the previous studies using MATLAB R2018a in the form of tables and graphs. Additionally, we presented a new version of the beta distribution and acquired some of its characteristics as an application in statistics. The outcomes produced here are generic and can give known and novel results.
... The logarithmic mean (L mean ) of θ, φ ∈ R + , which is of interest in many fields such as engineering, statistics, geometry, and thermodynamics (for more details, see [1][2][3]), is defined in the following integral formula ...
... Following [1], the L mean θ, φ is continuous when θ = φ and homogeneous and symmetric in θ and φ. Further, for θ, φ ∈ R + , we have ...
... Recently, various studies and generalizations for the L mean have been presented by several researchers (see e.g., [1][2][3][4][5][6]). ...
This paper aimed to obtain generalizations of both the logarithmic mean () and the Euler's beta function (EBF), which we call the extended logarithmic mean () and the extended Euler's beta-logarithmic function (EEBLF), respectively. Also, we discussed various properties, including functional relations, inequalities, infinite sums, finite sums, integral formulas, and partial derivative representations, along with the Mellin transform for the EEBLF. Furthermore, we gave numerical comparisons between these generalizations and the previous studies using MATLAB R2018a in the form of tables and graphs. Additionally, we presented a new version of the beta distribution and acquired some of its characteristics as an application in statistics. The outcomes produced here are generic and can give known and novel results.
... First and second logarithmic mean regressions have values for p that are close to 1 3 respectively. Power mean regression of order 1 3 is also called Lorentz mean regression (ZMR). Theorem 12: (Power Mean Equivalence Theorem) Let b be the slope of a generalized least-squares regression line. ...
... The case of jbj = 1 plays a special role in the previous theorem; therefore it is given a name. It is known and observed here that the Lorentz mean or power mean of order 1 3 , approximates the logarithmic mean well [12]. The power mean of order 1 3 approximates both the second logarithmic mean and the geometrically weighted harmonic-geometric mean H 1=3 G 2=3 , usually written as HG 2 1=3 [3,13]. ...
... In the (s; p) plane, the geometric mean corresponds to (s; p) = ( 1; 0), the power mean of order 1 2 corresponds to (s; p) = 1 2 ; 1 2 , and the arithmetic mean corresponds to (2; 1). The linear approximation is again the single straight line which passes through these points. ...
The theory of generalized least-squares is reformulated here using the notion of generalized means. The generalized least-squares problem seeks a line which minimizes the average
generalized mean of the square deviations in x and y. The notion of a generalized mean is equivalent to the generating function concept of the previous papers but allows for a more
robust understanding and has an already existing literature.
Generalized means are applied to the task of constructing more examples, simplifying the theory, and further classifying generalized least-squares regressions.
Available at: https://academicworks.cuny.edu/kb_pubs/77/
... The study of the inequalities involving the classical means such as arithmetic mean A, geometric mean G, identric mean I and logarithmic mean L have been of the extensive interest for several authors, e.g., see [2,3,9,11,20,21,29,30,31,41]. ...
... Discrete Mathematics [28]. For the historical background and the generalization of these means we refer the reader to see, e.g, [3,9,17,20,21,26,29,30,31,32,33,41]. Connections of these means with the trigonometric or hyperbolic inequalities can be found in [5,27,28,31]. ...
In this paper authors establish the two sided inequalities for the following two new means As well as many other well known inequalities involving the identric mean I and the logarithmic mean are refined from the literature as an application.
... It plays an important role in engineering problems involving heat and mass transfer: see [2,Chapter 4] for more details. It has several characterizations [3] as follows: ...
... We have successfully constructed the logarithmic mean of positive invertible operators in two different schemes, which are the integral representation and the skewed mean iteration. These are generalizations for the logarithmic mean of positive numbers introduced by Carlson [3]. We close with two open questions. ...
As a generalization of the logarithmic mean of positive real numbers, we establish the logarithmic mean of two positive invertible operators with two different construction schemes using Bochner integral and the convergence of skewed mean iteration. Indeed, we see that two constructions are the same for a non-weighted version with the uniform probability. Furthermore, we study several fundamental properties for the logarithmic mean of two positive invertible operators, and also investigate the logarithmic mean under tolerance relation on the open convex cone of positive definite Hermitian matrices.
... 1. Introduction. The study of the inequalities involving the classical means such as arithmetic mean A, geometric mean G, identric mean I and logarithmic mean L have been of the extensive interest for several authors, e.g., see [2,3,9,11,21,22,30,31,32,40]. ...
... For the historical background and the generalization of these means we refer the reader to, e.g. [3,9,17,21,22,28,30,31,32,33,34,40]. Connections of these means and the trigonometric or hyperbolic inequalities are given in [5,27,29,32]. ...
In this paper we establish two sided inequalities for the following two new means X = X(a, b) = AeG/P-1, Y = Y (a, b) = GeL/A-1, where A, G, L and P are the arithmetic, geometric, logarithmic, and Seiffert means, respectively. As an application, we refine many other well known inequalities involving the identric mean I and the logarithmic mean L.
... The study of the inequalities involving the classical means such as arithmetic mean A, geometric mean G, identric mean I and logarithmic mean L have been of the extensive interest for several authors, e.g., see [2,3,9,11,20,21,29,30,31,41]. ...
... Discrete Mathematics [28]. For the historical background and the generalization of these means we refer the reader to see, e.g, [3,9,17,20,21,26,29,30,31,32,33,41]. Connections of these means with the trigonometric or hyperbolic inequalities can be found in [5,27,28,31]. ...
In this paper authors establish the two sided inequalities for the following two new means As well as many other well known inequalities involving the identric mean I and the logarithmic mean are refined from the literature as an application.
... Carlson [5] gives bounds for L(x, y) in terms of G(x, y) and A(x, y) as follows ...
... holds with p 2 if and only if λ p ( 2 e ) p and μ p 2 3 . In this paper, we give the following generalizations of (3) and (5), and in the process a simpler of (7). ...
In this paper, some new bounds for L^p(x, y) and I^p(x, y) in terms of A^p(x, y) and
G^p(x, y) are established.
... The study of the inequalities involving the classical means such as arithmetic mean A, geometric mean G, identric mean I and logarithmic mean L has been of the extensive interest for several authors, e.g., see [1,2,8,10,18,19,27,28,29,38]. ...
... Recently, Sándor [26] introduced a new mean Y (a, b) for two positive real a and b, which is defined by respectively. For the historical background and the generalization of these means we refer the reader to see, e.g, [2,8,15,18,19,24,27,28,29,30,31,38]. Connections of these means with the trigonometric or hyperbolic inequalities can be found in [4,25,26,29]. ...
In this paper author establishes the two sided inequalities for the following
S\'andor means and other
related means.
... Hence there exists a corresponding operator logarithmic mean. B. C. Carlson [6] has shown that the logarithmic mean on R + is the skewed iterated composition of the geometric and arithmetic mean. It then follows from the theory of operator means as developed in Section 6 of [11], particularly Lemma 6.1 and Theorem 6.2, that the operator logarithmic mean is the corresponding skewed iterated composition of the operator geometric and arithmetic means on Ω, the set of positive operators on a Hilbert space, where the limits are taken in the weak operator topology. ...
In this paper we study the problem of extending means to means of higher order. We show how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces. As a particular application, we consider the positive operators on a Hilbert space under the Thompson metric and show that the operator logarithmic mean admits extensions of all higher orders, thus providing a positive solution to a problem of Petz and Temesi.
... (2) can be decomposed by employing the logarithmic mean; for example, Carlson (1972) investigated and defined its properties. For instance, it can be defined for optimistic numbers, i.e., m and n as: where mand nare the two positive numbers. ...
Carbon dioxide emissions (CO2es) are presently a hot topic of worldwide concern. It is of great significance for lessening CO2es to wholly understand the transformation pattern of CO2es among countries, industries, and the main factors (i.e., emission effect, energy intensity, economic development, population size, carbon per unit of land, land per capita, and environmental impact per capita effects) influencing CO2es. Thus, to mitigate the country’s CO2es efficiently, it is necessary to determine the driving factors of its emissions and damage variations. For this, we use the logarithmic mean Divisia index method. This research decomposes the major two dimensions, such as carbon sources and carbon damage variations from 1986 to 2020, into eight factors. The results show that Pakistan’s CO2es increased continuously during the period, with an average annual growth rate of 4.76%. Growing the country’s CO2es over 1986–2020, the key influencing factors are economic development, population, and land, while energy intensity and emission factors are the main forces in mitigating CO2es. The carbon source and carbon damage dimensions reached 68.75 Mt and 208.56 Mt, respectively, which led to a rise in CO2e. The entire set of factors is averagely moving around the major outcomes that provide significant policy measures. Finally, to efficiently reduce CO2e, Pakistan should concentrate on specific industrial paths and implement challenging, comprehensive governance to attain a low-carbon chain throughout the process. Thus, based on empirical results, this research put forward policy suggestions for cleaner production to reduce CO2 emissions further, and environmental policies must be tailored to local conditions.
... if (Carlson, 1972). ...
... This chapter describes how we have to extend and adjust the typical formulation of the logmean function (see Carlson (1972)) to be valid for all possible input values that can appear in dynamic simulation models including heat exchangers. ...
To calculate the driving temperature difference between the hot and the cold side of a heat exchanger, the use of the logarithmic mean temperature difference is common practice. To provide high robustness in complex dynamic system models, a robust formulation of the logarithmic mean (logmean) function becomes vital. As the analytic definition of the logmean function naturally comes along with singularities and limitations for specific input conditions, it is essential to extend and modify it for heat exchanger modeling. This paper proposes how the logmean function can be extended to be valid in all four quadrants of the Cartesian coordinate system and how to bridge the resulting definition gaps. Special focus lies on the robust formulation in such a way that it can be easily handled by numerical solvers. This includes the numerical approximation of the logmean by use of its integral form by implicit ODE solvers with variable step width. Furthermore a way is presented to flatten the naturally steep gradients in the vicinity of the x- and y-axes without manipulating the function in the uncritical regions. All the modifications on the logmean are finally applied in a simple simulation model written in the object-oriented programming language Modelica to examine the robustness of the approach.
... is well known and has been extensively studied. It is called logarithmic mean [see Carlson (1972)]. It separates the geometric and arithmetic means, provides a "true" mean: ...
In this paper, we consider the response of the oil-futures option market to the onset of severe conditions in the aftermath of Feb. 15, 2020. Motivated in part by the decline of the WTI futures contract into negative territory on April 20, 2020, for the derivative market on oil futures we consider an analytical contrast between the traditional Black model and its long-ago predecessor, the Bachelier model. Under 2020 crash conditions, the Bachelier model performs better than Black, displaying a significantly flatter vol smile. Based in part on previous published research for short-dated maturities , the rationale for this difference is built on the contrast between between implied Black and Bachelier volatilities. Other than for extreme strikes and high Black vols, we show that the rapport works well in a wider range of maturities and volatilities. Using options data over the year 2020, we explore a notion of normalized strike to measure quantitatively the vol skew.
... There are many interesting inequalities for these means, see for example, [3,6,8,10,13,14,15,20,22,25,26,29]. Here we mention several chains of inequalities for these means. ...
In this paper, we establish an interesting chain of sharp inequalities involving Toader-Qi mean, exponential mean, logarithmic mean, arithmetic mean and geometric mean. This greatly improves some existing results.
... Additionally, the above equation can be decomposed using the logarithmic mean (for a detailed description of its properties, see Carlson, 1972), defined for positive numbers x and y as: ...
We analyze the driving factors of CO2 emissions and energy intensity in Colombia during 1971–2017 and 1975–2016, respectively. We apply a factorial additive decomposition for CO2 emissions, starting from the Kaya identity, using the logarithmic mean Divisia index method. The increase in emissions is mainly explained by the affluence and population effects, but is partially offset by the energy intensity effect and, to a lesser extent, the carbonization effect. We then analyze the driving factors of energy intensity with a multiplicative decomposition. We first transform final energy into its total primary energy requirements. The decrease in energy intensity is mainly due to the reduction in sectoral energy intensity and, to a lesser extent, to structural change. We analyze the contribution to both effects of the different sectors considered and relate them with the structural changes of the Colombian economy and the policies applied. The most important contributions to sectoral energy intensity reduction are the efficiency improvements in the transport and industry sectors, while the decrease in the share of industry is the most relevant component explaining the reduction of the structural change effect. The results provide useful information for the analysis and design of energy and environmental policies.
... The crucial idea behind finding a numerically tractable version of an entropy conservative surface flux is the evaluation of its components at various means states between u L and u R . These mean state expressions can take on incredibly complex forms that depend on the arithmetic mean, the product of arithmetic means or more uncommon quantities like the logarithmic mean (Carlson, 1972). We have already introduced notation for the arithmetic mean, e.g. ...
Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Concerning robustness, DG has undergone an extensive transformation over the past seven years into its modern form that provides statements on solution boundedness for linear and nonlinear problems. This chapter takes a constructive approach to introduce a modern incarnation of the DG spectral element method for the compressible Navier-Stokes equations in a three-dimensional curvilinear context. The groundwork of the numerical scheme comes from classic principles of spectral methods including polynomial approximations and Gauss-type quadratures. We identify aliasing as one underlying cause of the robustness issues for classical DG spectral methods. Removing said aliasing errors requires a particular differentiation matrix and careful discretization of the advective flux terms in the governing equations.
... As Carlson remarked in [2], the logarithmic mean can be rewritten as ...
In this paper, we deal with the problem of finding the best possible bounds for the first Seiffert mean in terms of the geometric combination of logarithmic and the Neuman–Sándor means, and in terms of the geometric combination of logarithmic and the second Seiffert means.
... The above proposition gives comparisons between Heinz and Heron means. Some of the previous inequalities have been obtained by Bhatia in [4] and Carlson in [5]. On the other hand, in [18] Leach and Sholander showed that ...
In this paper we establish different refinements and applications of the Hermite–Hadamard inequality for convex functions in the context of NPC global spaces.
... Remark. Carlson [1] and Lin [2] gave some inequalities on mean and logarithmic mean. ...
This is corrigendum to New Inequalities on Hoomogeneous FunctionsDOI : http://dx.doi.org/10.22342/jims.21.1.201.71-72
... For two positive real numbers a and b, the Arithmetic, Geometric, Harmonic, Logarithmic, Identric, Lehmer, Modified Alzer, Power mean of order t ∈ R, and the weighted Power mean are respectively defined by , a = b, L p (a, b) = a p + b p a p−1 + b p−1 , p > 0, connection with elementary functions and with each other, we refer the reader to [2,3,11,12,13,14,16,15]. For the following definition see [5]. ...
Motivated by the work of Anderson, Vamanamurthy and Vuorinen \cite{avv}, in this paper authors study the log-convexity and log-concavity of Power mean, Identric mean, weighted Power mean, Lehmer mean, Modified Alzer mean, and establish the relation of these means with each other.
... This, with the known inequality < (1/3) + (2/3) (see, e.g., [8,9]), justifies our claim. ...
We investigate some results about mean-inequalities involving a large number of bivariate means. As application, we derive a lot of inequalities between four or more means among the standard means known in the literature.
... The first inequality of (2) was proved in 1983 by Leach and Sholander [28], whereas the second inequality of (2) was obtained in 1972 by Carlson [5] (see also Burk [4]) and earlier also by Pólya and Szegő [45]. Finally, let us note that some further refinements of both estimates are due to Chu and Long [6,29], Leach and Sholander [26,27], Matejíčka [35], Qian and Zheng [46], Sándor [47,48] and references therein, among others. ...
We provide a survey of several results on functional inequalities stemming from inequalities between classical means. Further, we recall a few problems in this field which according to the best of author’s knowledge remain open. Last section of this paper is devoted to a new, more general functional inequality and a joint generalization of several earlier results is obtained.
In this paper, by using the monotone form of L’Hospital’s rule and a criterion for the monotonicity of quotient of two power series we present some sharp bounds for a generalized logarithmic mean and Heinz mean by weighted means of harmonic mean, geometric mean, arithmetic mean, two power means and . Operator versions of these inequalities are obtained except for those related to the quadratic mean .
Consider the identric mean , the logarithmic mean two trigonometric means defined by H. J. Seiffert and denoted by and and the hyperbolic mean defined by E. Neuman and J. Sándor. There are a number of known inequalities between these means and some power means We add to these inequalities some new results obtaining the following chain of inequalities
We present a simple method for establishing several inequalities relating to logarithmic means.
As is known, all geo-logarithmic indices enjoy the axiomatic properties of being proportional, commensurable and homogeneous, together with their cofactors (Martini 1992a). Geologarithmic price indices satisfying the axioms of monotonicity, basis reversibility and factor reversibility have been investigated by Marco Fattore (2010), who has shown that the superlative Fisher price index does not belong to this family of indices. In this article, we discuss geo-logarithmic price indices with reference to the Laspeyres-Paasche bounding test and we propose a modification of the considered index family that satisfies this test. We also modify the structure of geo-logarithmic indices by using an additional parameter and, following the economic approach, we list superlative price index formulas that are members of the considered price index family. We obtain a special subfamily that approximates superlative price indices and includes the Fisher, Walsh and Sato-Vartia price indices.
In this paper, we obtain new inequalities for the logarithmic mean and the complete elliptic integral of the first kind. In order to prove the inequalities, we use the monotonicity property of sequences defined by these functions. Additionally, we apply our approach to previous studies. As a result, we get refinements of known inequalities.
Abstract In the article, we prove that the double inequality α L ( a , b ) + ( 1 − α ) T ( a , b ) < NS ( a , b ) < β L ( a , b ) + ( 1 − β ) T ( a , b ) holds for a , b > 0 with a ≠ b if and only if α ≥ 1 / 4 and β ≤ 1 − π / [ 4 log ( 1 + 2 ) ] , where NS ( a , b ) , L ( a , b ) L(a,b) and T ( a , b ) T(a,b) denote the Neuman-Sándor, logarithmic and second Seiffert means of two positive numbers a and b, respectively.
Abstract.
In this paper, optimal weighted geometric mean bounds of centroidal and harmonic means for convex combination of logarithmic and identric means are proved. We find the greatest value gamma(alpha) and the least value beta(alpha) for each 0<alpha< 1 such that the double inequality:
C^gamma(alpha)(a,b)H^(1-gamma(alpha))(a,b) < alphaL(a,b)+(1-alpha)I(a, b) <C^beta(alpha)(a,b)H^(1-beta(alpha))(a,b) holds for all a,b > 0 with a<> b. Here, C(a,b), H(a,b), L(a,b) and I(a,b) denote centroidal, harmonic, logarithmic and identric means of two positive numbers a and b, respectively.
101.07 Cauchy's mean value theorem meets the logarithmic mean - Volume 101 Issue 550 - Peter R. Mercer
By now we know Euler’s number quite well. In this chapter we define the exponential function for any x ∈ R, and its inverse the natural logarithmic function ln(x), for x > 0. (In the first section of the chapter we take a concise approach to the exponential function; in the second section we do things carefully.) These functions enable us to extend many of our previous results to allow for real exponents. For example, we obtain the Power Rule for real exponents, we extend Bernoulli’s Inequality, and we obtain a more strapping version of the AGM Inequality. We also meet the Logarithmic Mean, the Harmonic series and its close relatives the Alternating Harmonic series and p-series, and Euler’s constant \upgamma.
By way of the Fundamental Theorem of Calculus (Theorem 10. 1), many properties of integrals come from properties of derivatives and vice-versa. For example, the most basic technique of integration is to recognize the integrand as the derivative of some particular function. We saw a few examples of this sort of thing in the previous chapter. Here we focus on arguably the next two most important techniques of integration: u-Substitution which comes from the Chain Rule for derivatives, and Integration by Parts which comes from the Product Rule for derivatives.
In this paper, we obtain some new inequalities in the exponential form for the whole
of the triples about the four functions {1, (sinh t)/t, exp (t coth t – 1), cosh t}. Then we
generalize some well-known inequalities for the arithmetic, geometric, logarithmic,
and identric means to obtain analogous inequalities for their pth powers, where p > 0.
Given a system with a finite heat capacity and a heat reservoir, and two values of initial temperatures, T+ and T−(<T+), we inquire, in which case is the optimal work extraction larger: when the reservoir is an infinite source at T+ and the system is a sink at T−, or, when the reservoir is an infinite sink at T− and the system acts as a source at T+? It is found that in order to compare the total extracted work, and the corresponding efficiency in the two cases, we need to consider three regimes as suggested by an inequality, the so-called arithmetic mean-geometric mean inequality, involving the arithmetic and the geometric means of the two temperature values T+ and T−. In each of these regimes, the efficiency at total work obeys certain universal bounds, given only in terms of the ratio of initial temperatures. The general theoretical results are exemplified for thermodynamic systems for which internal energy and temperature are power laws of the entropy. The conclusions may serve as benchmarks in the design of heat engines, where we can choose the nature of the finite system, so as to tune the total extractable work and/or the corresponding efficiency.
For many consumer-intensive (i.e., business-to-consumer) services, delivering memorable customer experiences is a source of competitive advantage. Yet there are few guidelines available for designing service encounters with a focus on customer satisfaction. In this paper, we show how experiential services should be sequenced and timed to maximize the satisfaction of customers who are subject to memory decay and acclimation. We find that memory decay favors positioning the highest service level near the end, whereas acclimation favors maximizing the gradient of service level. Together, they maximize the gradient of service level near the end. Although memory decay and acclimation lead to the same design individually, they can act as opposing forces when considered jointly. Overall, our analysis suggests that short experiences should have activities scheduled as a crescendo and duration allocated primarily to the activities with the highest service levels, whereas long experiences should have activities scheduled in a U-shaped fashion and duration allocated primarily to activities with the lowest service level so as to ensure a steep gradient at the end.
In the article, we prove that the double inequality lambda root L(a,b)I(a,b) < TQ(a,b) < mu root L(a,b)I(a,b) holds for all a,b > 0 with a not equal b if and only if lambda <= root e/pi and mu >= 1, and give an affirmative answer to the conjecture proposed by Yang in [39], where L(a,b) = (b-a)/(logb-loga), I(a,b) = (b(b)/a(a))(1/(b-a))/e and TQ(a,b) = 2/pi integral(pi/2)(0) a(cos2 theta)b(sin2 theta) d theta are respectively the logarithmic, identric and Toader-Qi means of a and b.
In the article, we present several sharp bounds for the modified Bessel function of the first kind I0(t)=∑n=0∞t2n22n(n!)2 and the Toader-Qi mean TQ(a,b)=2π∫0π/2acos2θbsin2θdθ for all t>0 and a,b>0 with a≠b.
This paper describes interconvertible rules between an aggregative index like the Laspeyres index and a log-change index like the Törnqvist index. Thus we can compare an aggregative index with a log-change index in the same form. Using these rules, we formulate the logarithmic dierence between the Laspeyres price index and the Törnqvist price index. One of the rules may be combined with another. By using these combined rules, we can change from given weights to other weights in an aggregative index (or a log-change index) of which the value is invariable.
For p ∈ R the p-th one-parameter mean Jp(a, b) of two positive numbers a and b with a ≠ b is defined by In this article, we answer the question: What are the greatest value α and the least value β, such that the double inequality Jα(a, b) < I(a, b) < Jβ(a, b) holds for all a, b > 0 with a ≠ b? Here denotes the identric mean of a and b.
The paper deals with the generalized Gauss composition of arbitrary means. We give sufficient conditions for the existence of this generalized Gauss composition. Finally, we show that these conditions cannot be improved or changed.
We find the greatest values α1 and α2, and the least values β 1 and β 2 such that the inequalities α1C(a, b)+(1-α1)H(a, b)< L(a, b) < β1C(a, b)+(1-β1)H(a, b) and α2C(a; b)+(1- α2)H(a, b)< I(a, b) < β2C(a, b) + (1-β2)H(a; b) hold for all a, b > 0 with a 6= b. Here, C(a, b), H(a, b), L(a, b), and I(a, b) are the centroidal, harmonic, logarithmic, and identric means of two positive numbers a and b, respectively.
The invariance equality L ○ (M, N) = L, where L is the logarithmic mean, and where the unsymmetric compound means M = A ○ (P1, G), N = A ○ (P2, G) are built with the arithmetic A, geometric G, and projective means P1, P2, is called "Carlson's log" and is important in iteration of means. In the present paper we present effective and simple 1-parameter families of unsymmetric means Mt, Nt : (0,∞)2 → (0,∞) such that, for all t ∈ (-1, 1), L ○ (Mt, Nt) = L and M = M 1/2, N = N 1/2. Existence of elementary (simple) symmetric means M and N such that L ○ (M, N) = L and M ≠ L is posed as an open problem.