Monotone Convex Functions Research Articles

Equality cases in monotonicity of quasi-entropies, Lieb’s concavity and Ando’s convexity

We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let Φ:B(H)→B(K) be a trace-preserving map such that Φ* is a Schwarz map. When f is an operator monotone or operator convex function on [0, ∞), we present several equivalent conditions for the equality SfK(Φ(ρ)‖Φ(σ))=SfΦ*(K)(ρ‖σ) to hold for given positive operators ρ, σ on H and K∈B(K). The conditions include equality cases in the monotonicity versions of Lieb’s concavity and Ando’s convexity theorems. Specializing the map Φ we have equivalent conditions for equality cases in Lieb’s concavity and Ando’s convexity. Similar equality conditions are discussed also for monotone metrics and χ2-divergences. We further consider some types of linear preserver problems for those quantum information quantities.

Upper and lower bounds for an integral transform of positive operators in Hilbert spaces with applications

For a continuous and positive function \(w(\lambda)\), \(\lambda>0\) and a positive measure \(\mu\) on \((0,\infty )\) we consider the following integral transform \[ \mathcal{D}( w,\mu ) ( T) :=\int_{0}^{\infty }w(\lambda) (\lambda +T)^{-1} d\mu ( \lambda ) , \] where the integral is assumed to exist for any positive operator \(T\) on a complex Hilbert space \(H\). In this paper we obtain several upper and lower bounds for the difference \(\mathcal{D}( w,\mu ) ( A) -\mathcal{D}( w,\mu ) ( B)\) under certain assumptions for the operators \(A\) and \(B\). Some natural applications for operator monotone and operator convex functions are also given.

A Brief Review of Operator Monotone and Operator Convex Functions

This article is about transient information regarding the functions that satisfy the conditions of monotonicity and convexity on a set, onset of matrices of particular patterns with finite or infinite order called operator monotone or operator convex functions. Data is gathered by investigating numerous existing articles, review papers, publications that show wide scattering in multiple dimensions according to the behavior and prolific results. This work emphasizes not only the origin, key results, and applications of convexity and monotonicity but also their extensions in different directions. It also takes us to the current research scenario in relating fields and may provide an opportunity to express some new ideas, methods, or concepts.

Hardy-Type Inequalities for an Extension of the Riemann- Liouville Fractional Derivative Operators

In this paper we present variety of Hardy-type inequalities and their refinements for an extension of Riemann-Liouville fractional derivative operators. Moreover, we use an extension of extended Riemann-Liouville fractional derivative and modified extension of Riemann-Liouville fractional derivative using convex and monotone convex functions. Furthermore, mean value theorems and n-exponential convexity of the related functionals is discussed.

Some inequalities involving operator means and monotone convex functions

Some inequalities involving operator means and monotone convex functions

Norm inequalities involving operator monotone functions

ABSTRACT In this paper, we show an analogue inequality of geometric concavity property for operator monotone functions involving Kantorovich constant. This inequality provides some unitarily norm inequalities involving operator monotone and operator convex functions.

The General Least Square Deviation OWA Operator Problem

A crucial issue in applying the ordered weighted averaging (OWA) operator for decision making is the determination of the associated weights. This paper proposes a general least convex deviation model for OWA operators which attempts to obtain the desired OWA weight vector under a given orness level to minimize the least convex deviation after monotone convex function transformation of absolute deviation. The model includes the least square deviation (LSD) OWA operators model suggested by Wang, Luo and Liu in Computers & Industrial Engineering, 2007, as a special class. We completely prove this constrained optimization problem analytically. Using this result, we also give solution of LSD model suggested by Wang, Luo and Liu as a function of n and α completely. We reconsider two numerical examples that Wang, Luo and Liu, 2007 and Sang and Liu, Fuzzy Sets and Systems, 2014, showed and consider another different type of the model to illustrate our results.

Weighted Hardy-type inequalities involving convex function for fractional calculus operators

The aim of this paper is to establish some new weighted Hardy-type inequalities involving convex and monotone convex functions using Hilfer fractional derivative and fractional integral operator with generalized Mittag-Leffler function in its kernel. We also discuss one dimensional cases of our related results. As a special case of our general results we obtain the results of Iqbal et al. (2017). Moreover, the refinement of Hardy-type inequalities for Hilfer fractional derivative is also included.

More on operator monotone and operator convex functions of several variables

Let C1,C2,…,Ck be positive matrices in Mn and f be a continuous real-valued function on [0,∞). In addition, consider Φ as a positive linear functional on Mn and defineϕ(t1,t2,t3,…,tk)=Φ(f(t1C1+t2C2+t3C3+…+tkCk)), as a k variables continuous function on [0,∞)×…×[0,∞). In this paper, we show that if f is an operator convex function of order mn, then ϕ is a k variables operator convex function of order (n1,…,nk) such that m=n1n2…nk. Also, if f is an operator monotone function of order nk+1, then ϕ is a k variables operator monotone function of order n. In particular, if f is a non-negative operator decreasing function on [0,∞), then the function t→Φ(f(A+tB)) is an operator decreasing and can be written as a Laplace transform of a positive measure.

Inertia of some special matrices

There have been several articles in literature that study the spectral behaviour of special classes of matrices such as the matrices based on power function given by the matrices , the matrices for positive values of r and positive real numbers . Bhatia and Jain in 2015 and Dyn, Goodman and Micchelli in 1986 have studied the spectral behaviour of and , respectively, for all real values of r. The power function is operator monotone when and operator convex when . It is natural to study the inertia of all these matrices when the power function is replaced by any operator monotone or operator convex function. In the present work, inertia of the matrices and is discussed for non-negative operator monotone and operator convex function f, which further leads to many known and new results.

On some Hardy-type inequalities for fractional calculus operators

In this article we present applications of Hardy-type and refined Hardy-type inequalities for a generalized fractional integral operator involving the Mittag-Leffler function in its kernel and for the Hilfer fractional derivative using convex and monotone convex functions.

On operator monotone and operator convex functions

We establish monotonicity and convexity criteria for a continuous function f: R+ → R with respect to any C*-algebra. We obtain an estimate for the measure of noncompactness of the difference of products of the elements of a W*-algebra. We also give a commutativity criterion for a positive τ-measurable operator and a positive operator from a von Neumann algebra.

Applications of refined Hardy-type inequalities

This paper is to provide the broad range of Hardy-type inequalities and their refinements for linear differential operator, Widder’s derivative and more generalized fractional integral operator using convex and monotone convex functions. As special cases we give results for Saigo, Riemann-Liouville and Erdelyi-Kober fractional integral operators.

A General Model of Random Variation

A statistical distribution of a random variable is uniquely represented by its normal-based quantile function. For a symmetrical distribution it is S-shaped (for negative kurtosis) and inverted S-shaped (otherwise). As skewness departs from zero, the quantile function gradually transforms into a monotone convex function (positive skewness) or concave function (otherwise). Recently, a new general modeling platform has been introduced, response modeling methodology, which delivers good representation to monotone convex relationships due to its unique “continuous monotone convexity” property. In this article, this property is exploited to model the normal-based quantile function, and explored using a set of 27 distributions.

A sharp bound for the ergodic distribution of an inventory control model under the assumption that demands and inter-arrival times are dependent

In this study, a stochastic process which represents a single-item inventory control model with (s,S)-type policy is constructed when the demands of a costumer are dependent on the inter-arrival times between consecutive arrivals. Under the assumption that the demands can be expressed as a monotone convex function of the inter-arrival times, it is proved that this process is ergodic, and closed form of the ergodic distribution is given. Moreover, a sharp lower bound for this distribution is obtained.

Maximum Lebesgue extension of monotone convex functions

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a “nice” dual representation of the function.

On the Lebesgue property of monotone convex functions

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous linear functionals. This generalizes and unifies several recent results obtained in the context of convex risk measures.

Sum Formula for Maximal Abstract Monotonicity and Abstract Rockafellar’s Surjectivity Theorem

In this paper, we present an example in which the sum of two maximal abstract monotone operators is maximal. Also, we shall show that the necessary condition for Rockafellar's surjectivity which was obtained in ((19), Theorem 4.3) can be sufficient. Abstract convexity has found many applications in the study of problems of mathematical analysis and optimization. Abstract convexity has mainly been used for the study of point-to-point functions. Examples of its use in the analysis of multifunctions can be found in (3, 13, 14, 24). Several approaches to the theory of monotone multifunctions have established links between maximal monotone multifunctions and convex functions (see

Weighted Hardy-type inequalities for monotone convex functions with some applications

In this paper, we establish some new refined weighted Hardy-type inequalities in- volving monotone convex functions. We give the results for some special kernels of Riemann- Liouville and Weyl's operators as applications. Also we discuss some related dual cases. At the end, we prove some refined G. H. Hardy-type inequalities for different kinds of fractional integrals and fractional derivatives.

Convex envelopes of products of convex and component-wise concave functions

In this paper, we consider functions of the form $${\phi(x,y)=f(x)g(y)}$$ over a box, where $${f(x), x\in {\mathbb R}}$$ is a nonnegative monotone convex function with a power or an exponential form, and $${g(y), y\in {\mathbb R}^n}$$ is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.