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The Weighted Logarithmic Mean
... It seems that there are not familiar weighted means except the power and Heron means. Some researchers discussed the weighted logarithmic mean in their own way in [9,10,13,14]. Recently, based on the Hermite-Hadamard inequality for convex functions, Pal, Singh, Moslehian and Aujla [10] introduced the weighted logarithmic ...
... We can consider plural weighted means from one symmetric mean. In fact, the weighted logarithmic mean is defined by several ways in [9,10,13,14]. Moreover, in [1,2,11,16], they discussed the algorithms to make weighted operator means from a given operator mean. ...
... The following remark may be of interest. , which are the symmetric multivariate means introduced in the literature, see [12] and [20]. ...
Recently, the so-called Hermite-Hadamard inequality for (operator) convex functions with one variable has known extensive several developments by virtue of its nice properties and various applications. The fundamental target of this paper is to investigate a weighted variant of Hermite-Hadamard inequality in multiple variables that extends the univariate case. As an application, we introduce some weighted multivariate means extending certain bivariate means known in the literature.
... Basic equation of piezo-electric materials is also used. Further in case of generation of energy at micro level various vital hybrid techniques like Taguchi and heterogeneous 3-D has been used for optimization, simulation, modelling and fabrication in case of low-level vibration applications [54,55,56,57,58]. ...
In this work, the different kinds of fluid flow interactions with piezo smart materials have been discussed for energy harvesting. The present work has been classified into the following categories: (i) experimental investigations (ii) simulation and (iii) mathematical modelling. In section (i) different experimental set-ups such as harvesting of energy with the help of vortex flow, turbulent flow, cross flow, flow in an open channel and closed channel and flow through nozzles have been examined. In section (ii) simulations studies performed with different tools/software like ANSYS/fluent, COSMOL etc. have been detailed. Lastly, in section (iii) different mathematical equations such as Navier-Stokes equation of motion, Continuity equation, finite element method, numerical methods, transport equations, Bernoulli equation, equation of linear elasticity, fluid structural equations, piezoelectric equations and coupled-wave equations are described for generation of energy with fluid’s interaction. The present work has to fulfil two aims: (i) active engineers can choose the best appropriate methodology for their work in the same field (ii) researchers can know about how the area of energy harvesting has been grown in various decades, what are the practical application of this field with real life and the literature gaps of the field.
... whereas Neuman [112] has also provided an alternative integral representation: ...
In this chapter, we present some other types of applications of the aforementioned SOC-functions, SOC-convexity, and SOC-monotonicity. These include so-called SOC means, SOC weighted means, and a few SOC trace versions of Young, Hölder, Minkowski inequalities, and Powers–Størmer’s inequality. We believe that these results will be helpful in convergence analysis of optimizations involved with SOC. Many materials of this chapter are extracted from [36, 77, 78], the readers can look into them for more details.
... whereas Neuman [112] has also provided an alternative integral representation: ...
In this chapter, we introduce the SOC-convexity and SOC-monotonicity which are natural extensions of traditional convexity and monotonicity. These kinds of SOC-convex and SOC-monotone functions are also parallel to matrix-convex and matrix-monotone functions, see [21, 74]. We start with studying the SOC-convexity and SOC-monotonicity for some simple functions, e.g., \(f(t)=t^2, t^3, 1/t, t^{1/2}, |t|\), and \([t]_{+}\). Then, we explore characterizations of SOC-convex and SOC-monotone functions.
... whereas Neuman [112] has also provided an alternative integral representation: ...
During the past two decades, there have been active research for second-order cone programs (SOCPs) and second-order cone complementarity problems (SOCCPs). Various methods had been proposed which include the interior-point methods [1, 102, 109, 123, 146], the smoothing Newton methods [51, 63, 71], the semismooth Newton methods [86, 120], and the merit function methods [43, 48]. All of these methods are proposed by using some SOC complementarity function or merit function to reformulate the KKT optimality conditions as a nonsmooth (or smoothing) system of equations or an unconstrained minimization problem. In fact, such SOC complementarity functions or merit functions are closely connected to so-called SOC functions. In other words, studying SOC functions is crucial to dealing with SOCP and SOCCP, which is the main target of this chapter.
The extended power difference mean \(f_{a,b}(t):={b\over a}{{t^a-1}\over {t^b-1}}\) \((a,b\in {\mathbb {R}})\) is investigated in this paper. We show some Thompson metric inequalities involving \(f_{a,b}\) and Tsallis relative operator entropy. We also discuss the behavior of the bivariate function defined as the perspective map for \(f_{a,b}\). Finally, the relationship beween \(f_{a,b}\) and the weighted logarithmic mean is studied.
The Hermite–Hadamard inequalities attract many researchers and are useful theoretical point of view as well as in practical purposes. Some properties, extensions, refinements, reverses and applications of these inequalities have been carried out recently. In this paper, we investigate some refinements and reverses for the Hermite–Hadamard inequalities when the integrand map is operator convex (resp. operator concave) in multiple operator arguments. As application, some inequalities for multivariate operator means are provided.
Using the integral representation of logarithmic mean, we define the logarithmic mean of multiple accretive matrices. When the number of matrices is two, it coincides with the recent studies carried out by Tan and Xie (Filomat 33:4747–4752, 2019). Several inequalites are presented along with the studies.
In this paper, we define various means associated with Lorentz cones (also known as second-order cones), which are new concepts and natural extensions of traditional arithmetic mean, harmonic mean, and geometric mean, logarithmic mean. Based on these means defined on the Lorentz cone, some inequalities and trace inequalities are established.
The problem of determining the moments and the Fourier transforms of B-splines with arbitrary knots is considered. There exists a simple connection between the moments of such splines and the so-called extended Stirling numbers of the second kind which are defined in section 2. Some recurrence relations for the moments of B-splines with arbitrary knots are given in section 3. In the case of equidistant knots we have also further recurrences. For the forward, central and perfect B-splines the explicit formulas for the moments are given in section 3. The Fourier transforms of B-splines is treated in section 4. The final section is devoted to so-called Stieltjes series connected with the nonnegative weight function w(x) and such that in some closed interval [a, b]. It is proved that such series for the particular values of the independent variable may be expressed by the finite sums which contain the nodes and coefficients of the optimal (in the Davies sense) quadrature formulas.
This paper is devoted to the study of some properties of the complete symmetric functions. These functions play an important role in the theory of partitions and in the combinatorics as well. Among other things, the representation formulas as well as the recurrence formulas and inequalities involving functions under discussion are given. Some applications are also included.
1. The weights w1,…, wn associated with a mean value M(x1,…, xn) are defined as partial derivatives of M and identified for various homogeneous means. 2. We prove Theorem 1: If a1,…, an are real and x → + ∞, then M(a1+x,…,an+x)=x+Σwiai+ , provided only that M satisfies very general hypotheses; in particular this asymptotic formula (originally discovered by L. Hoehn and I. Niven, Math. Mag., 58 (1985), 151–156, in special cases) is valid for power means, Dresher means, mixed means, and hypergeometric means. 3. We prove Theorem 2: If x1, x2,… is an infinite sequence of positive numbers lying in a finite interval [0 < α ⩽ xi ⩽ β] and if the arithmetic mean of the first n values approaches a limit as n → ∞, then the hypergeometric mean approaches the same limit. (This theorem was first established for the special case of the logarithmic mean of any order by A. O. Pittenger, J. Math. Anal. Appl., 123 (1987), 281–291.)
The present paper was written in 1945 and completed by 1947 (see the abstract [3]) but for no good reason has so far not been published. It appears now in a somewhat revised and improved form.