Gini Means Research Articles

Local and global Hölder- and Minkowski-type inequalities for nonsymmetric generalized Bajraktarević means

The aim of this paper is to investigate inequalities that are analogous to the Minkowski and Hölder inequalities by replacing the addition and the multiplication by a more general operation, and instead of using power means, generalized Bajraktarević means are considered, in particular, Gini means. A further aim is to introduce the concept of local and global validity of such inequalities and to characterize them in both senses.

Estimating the Hardy constant of nonconcave Gini means

Estimating the Hardy constant of nonconcave Gini means

Characterizations of the equality of two-variable generalized quasiarithmetic means

This paper is motivated by an astonishing result of H. Alzer and S. Ruscheweyh published in 2001, which states that the intersection of the classes two-variable Gini means and Stolarsky means is equal to the class of two-variable power means. The two-variable Gini and Stolarsky means form two-parameter classes of means expressed in terms of power functions. They can naturally be generalized in terms of the so-called Bajraktarević and Cauchy means. Our aim is to show that the intersection of these two classes of functional means, under high-order differentiability assumptions, is equal to the class of two-variable quasiarithmetic means.

WEAKENING OF THE HARDY PROPERTY FOR MEANS

The aim of this paper is to find a broad family of means defined on a subinterval of $I\subset [0,+\infty )$ such that $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\mathscr{M}(a_{1},\ldots ,a_{n})<+\infty \quad \text{for all }a\in \ell _{1}(I).\end{eqnarray}$$ Equivalently, the averaging operator $(a_{1},\,a_{2},a_{3}\,,\ldots )\mapsto (a_{1},\,\mathscr{M}(a_{1},a_{2}),\,\mathscr{M}(a_{1},a_{2},a_{3}),\ldots )$ is a selfmapping of $\ell _{1}(I)$. This property is closely related to the so-called Hardy inequality for means (which additionally requires boundedness of this operator). We prove that these two properties are equivalent in a broad family of so-called Gini means. Moreover, we show that this is not the case for quasi-arithmetic means, that is functions $f^{-1}(\sum f(a_{i})/n)$, where $f:I\rightarrow \mathbb{R}$ is continuous and strictly monotone, $n\in \mathbb{N}$ and $a\in I^{n}$. However, the weak Hardy property is localisable for this family.

Interval-type theorems concerning means

Abstract Each family ℳ of means has a natural, partial order (point-wise order), that is M ≤ N iff M(x) ≤ N(x) for all admissible x. In this setting we can introduce the notion of interval-type set (a subset ℐ ⊂ℳ such that whenever M ≤ P ≤ N for some M, N ∈ℐ and P ∈ℳ then P ∈ℐ). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. In the present paper we consider this property for Gini means and Hardy means. Moreover, some results concerning L ∞ metric among (abstract) means will be obtained.

Weak monotonicity of Lehmer and Gini means

We analyze directional monotonicity of several mixture functions in the direction (1,1…,1), called weak monotonicity. Our particular focus is on power weighting functions and the special cases of Lehmer and Gini means. We establish limits on the number of arguments of these means for which they are weakly monotone. These bounds significantly improve the earlier results and hence increase the range of applicability of Gini and Lehmer means. We also discuss the case of affine weighting functions and find the smallest constant which ensures weak monotonicity of such mixture functions.

On negative results concerning Hardy means

In the present paper we are going to prove some necessary condition for a mean to be Hardy. This condition is then applied to completely characterize the Hardy property among the Gini means.

On Gasparyan’s Inequality

We consider a new and simpler proof of an inequality of A.S. Gasparyan, which was originally derived in terms of complex algebraical objects --- multidimensional hyperdeterminants. Our proof is much simpler and use only standard technics such as mean inequalities. The main theorem on Gini means is essentially used. Some corollaries and problems are considered.

On the weak monotonicity of Gini means and other mixture functions

Weak monotonicity was recently proposed as a relaxation of the monotonicity condition for averaging aggregation, and weakly monotone functions were shown to have desirable properties when averaging data corrupted with outliers or noise. We extended the study of weakly monotone averages by analyzing their ϕ-transforms, and we established weak monotonicity of several classes of averaging functions, in particular Gini means and mixture operators. Mixture operators with Gaussian weighting functions were shown to be weakly monotone for a broad range of their parameters. This study assists in identifying averaging functions suitable for data analysis and image processing tasks in the presence of outliers.

Schur-geometric convexity of the generalized Gini-Heronian means involving three parameters

Abstract In this paper, we give a unified generalization of the Gini means and Heronian means. The Schur-geometric convexity of the generalized Gini-Heronian means are investigated. Our result generalizes an earlier result given by Shi et al. (J. Inequal. Appl. 2008:879273, 2008). At the end of the paper, two new inequalities related to the generalized Gini-Heronian means are established to illustrate the applicability of the given result. MSC:26D15, 26E60, 26A51.

Sixth-order modifications of Newton’s method based on Stolarsky and Gini means

In this article we present sixth order methods developed by extending third order methods of Herceg and Herceg (2013) for solving nonlinear equations. The methods require only four function evaluations per iteration. In this regard the efficiency index of our methods is 61/4≈1.56508. Considered methods are based on Stolarsky and Gini means and depend on two parameters. Sixth order convergence of considered methods is proved, and corresponding asymptotic error constants are expressed in terms of two parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, are included to demonstrate convergence and efficacy of our methods. For some combinations of parameter values, the new sixth order methods produce very good results on tested examples, compared to the results produced by some of the sixth order methods existing in the related literature.

SCHUR POWER CONVEXITY OF GINI MEANS

In this paper, the Schur convexity is generalized to Schur <TEX>$f$</TEX>-convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When <TEX>$f$</TEX> : <TEX>${\mathbb{R}}_+{\rightarrow}{\mathbb{R}}$</TEX> is defined as <TEX>$f(x)=(x^m-1)/m$</TEX> if <TEX>$m{\neq}0$</TEX> and <TEX>$f(x)$</TEX> = ln <TEX>$x$</TEX> if <TEX>$m=0$</TEX>, the necessary and sufficient conditions for <TEX>$f$</TEX>-convexity (is called Schur <TEX>$m$</TEX>-power convexity) of Gini means are given, which generalize and unify certain known results.

Third-order modifications of Newton’s method based on Stolarsky and Gini means

In this paper we consider third-order modifications of Newton’s method for solving nonlinear equations. Considered methods are based on Stolarsky and Gini means, Stolarsky (1975) [13], Stolarsky (1980) [14], Chen (2008) [12] and depend on two parameters. Some known methods are special cases of our methods, for example, the power mean Newton’s method Zhou (2007) [10], the arithmetic mean Newton’s method Weerakoon and Fernando (2000) [5], the harmonic mean Newton’s method, Özban (2004) [8] and the geometric mean Newton’s method, Lukić and Ralević (2008) [9]. Third order convergence of the considered methods is proved, and corresponding asymptotic error constants are expressed in terms of two parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, support the theoretical results. Some numerical tests were performed, and it was shown that our methods yield better numerical results (i.e. a smaller error |x4−α|) when compared to Halley, Euler, Hansen-Patrick, Ostrowski and inverse interpolation methods.

Sequences of Inequalities Among Differences of Gini Means and Divergence Measures

Abstract In 1938, Gini [3] studied a mean having two parameters. Later, many authors studied properties of this mean. In particular, it contains the famous means as harmonic, geometric, arithmetic, etc. Here we considered a sequence of inequalities arising due to particular values of each parameter of Gini’s mean. This sequence generates many nonnegative differences. Not all of them are convex. We have established new sequences of inequalities of these differences. Some refinement inequalities are also presented. Considering in terms of probability distributions these differences, we have made connections with some well known divergence measures.

Inequalities for hyperbolic functions

Several inequalities involving hyperbolic functions are derived. Some of them are obtained with the aid of Stolarsky and Gini means.

Logarithmic convexity of Gini means

In the paper, the monotonicity and logarithmic convexity of Gini means and related functions are investigated.

Inequalities between Arithmetic‐Geometric, Gini, and Toader Means

We find the greatest values p1, p2 and least values q1, q2 such that the double inequalities and hold for all a, b &gt; 0 with a ≠ b and present some new bounds for the complete elliptic integrals. Here M(a, b), T(a, b), and Sp(a, b) are the arithmetic‐geometric, Toader, and pth Gini means of two positive numbers a and b, respectively.

Minkowski-type inequalities for means generated by two functions and a measure

Given two continuous functions f; g : I ! R such that g is positive and f=g is strictly monotone, and a probability measure  on the Borel subsets of [0; 1], the two variable mean Mf;g; : I2 ! I is de ned by Mf;g;(x; y) := ( f g )..1 (�c �c 1 0 f ( tx + (1 .. t)y ) d(t) �c �c 1 0 g ( tx + (1 .. t)y ) d(t) ) (x; y 2 I): The aim of this paper is to study Minkowski-type inequalities for these means, i.e., to nd conditions for the generating functions f0; g0 : I0 ! R, f1; g1 : I1 ! R, . . . , fn; gn : In ! R, and for the measure  such that Mf0;g0;(x1 +


+ xn; y1 +


+ yn)  [] Mf1;g1;(x1; y1) +


+Mfn;gn;(xn; yn) holds for all x1; y1 2 I1, . . . , xn; yn 2 In with x1 +


+ xn; y1 +


+ yn 2 I0. The particular case when the generating functions are power functions, i.e., when the means are generalized Gini means is also investigated.

Necessary and Sufficient Conditions for Schur Geometrical Convexity of the Four‐Parameter Homogeneous Means

The necessary and sufficient conditions for Schur geometrical convexity of the four‐parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means.

Computer aided solution of the invariance equation for two-variable Gini means

Our aim is to solve the so-called invariance equation in the class of two-variable Gini means ${G_{p,q}:p,q\in\R}$, i.e., to find necessary and sufficient conditions on the 6 parameters $a,b,c,d,p,q$ such that the identity [G_{p,q}\big(G_{a,b}(x,y),G_{c,d}(x,y)\big)=G_{p,q}(x,y) \qquad (x,y \in \R_+)] be valid. We recall that, for $p\neq q$, the Gini mean $G_{p,q}$ is defined by [G_{p,q}(x,y):=(\dfrac{x^p+y^p}{x^q+y^q})^{\frac1{p-q}}\qquad (x,y \in \R_+).] The proof uses the computer algebra system Maple V Release 9 to compute a Taylor expansion up to 12th order, which enables us to describe all the cases of the equality.