Convex Order Research Articles

Starlikeness and Convexity of Generalized Bessel-Maitland Function

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order δ, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies.

ON THE GENERALIZATION OF SOME CLASSES OF CONVEX IN DIRECTION AND TYPICALLY REAL FUNCTIONS

In the article by M.O. Reade (Duke Math. Journal, 1956) using the condition |arg (f'(z)/ g'(z)) | ≤ γπ/2, where g(z) is a convex function,0≤γ≤1 , a class of functions close-to-convex (almost convex) of order γ is introduced. In our paper, we introduce a subclass of the class of close-to-convex (almost convex) order γ functions satisfying the condition |arg [(1-λzn ) f'(z)]| ≤ γπ/2, which, for different parameter values, gives a number of well-known subclasses of univalent (schlicht) functions. Based on this subclass, a class of close-to-starlike (almost star-shaped) functions is constructed, containing a number of subclasses that have been actively studied by many authors in recent years, as well as a classical class of typically real functions. For this classes exact theorems of distortion (growth) and radii of convexity (starlikeness) are obtained, generalizing previously known results. The case is also considered when the functions of the introduced classes have missing members in the power series expansion. The results obtained are accurate and not only generalize previously known results, but also reveal the properties of a number of new subclasses of univalent (schlicht) functions.

A Note on Convexity Properties for Gaussian Hypergeometric Function

Gaussian hypergeometric function has been investigated in the context of geometric function theory regarding many aspects. Obtaining univalence conditions for this function is a line of research followed by many scholars. In the present study, methods specific to the differential superordination theory are used for obtaining properties of the Gaussian hypergeometric function regarding convexity of order . Also, a necessary and sufficient condition is proved such that Gaussian hypergeometric function is a close-to-convex function. The applicability of the theoretical findings is demonstrated by a numerical example.

Convex and Lorenz orders under balance correction in nonlife insurance pricing: Review and new developments

By exploiting massive amounts of data, machine learning techniques provide actuaries with predictors exhibiting high correlation with claim frequencies and severities. However, these predictors generally fail to achieve financial equilibrium and thus do not qualify as pure premiums. Autocalibration effectively addresses this issue since it ensures that every group of policyholders paying the same premium is on average self-financing. Balance correction has been proposed as a way to make any candidate premium autocalibrated with the added advantage that it improves out-of-sample Bregman divergence and hence predictive Tweedie deviance. This paper proves that balance correction is also beneficial in terms of concentration curves and derives conditions ensuring that the initial predictor and its balance-corrected version are ordered in Lorenz order. Finally, criteria are proposed to rank the balance-corrected versions of two competing predictors in the convex order.

A New Look and Convergence Rate of Federated Multitask Learning With Laplacian Regularization.

Non-independent and identically distributed (non-IID) data distribution among clients is considered as the key factor that degrades the performance of federated learning (FL). Several approaches to handle non-IID data, such as personalized FL and federated multitask learning (FMTL), are of great interest to research communities. In this work, first, we formulate the FMTL problem using Laplacian regularization to explicitly leverage the relationships among the models of clients for multitask learning. Then, we introduce a new view of the FMTL problem, which, for the first time, shows that the formulated FMTL problem can be used for conventional FL and personalized FL. We also propose two algorithms FedU and decentrali- zed FedU ( dFedU ) to solve the formulated FMTL problem in communication-centralized and decentralized schemes, respectively. Theoretically, we prove that the convergence rates of both algorithms achieve linear speedup for strongly convex and sublinear speedup of order 1/2 for nonconvex objectives. Experimentally, we show that our algorithms outperform the conventional algorithm FedAvg, FedProx, SCAFFOLD, and AFL in FL settings, MOCHA in FMTL settings, as well as pFedMe and Per-FedAvg in personalized FL settings.

Geometric characterization of the generalized Lommel–Wright function in the open unit disc

The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function Jλ,μν,m(z):=Γm(λ+1)Γ(λ+μ+1)22λ+μz1−(ν/2)−λJλ,μν,m(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{J}_{\\lambda ,\\mu}^{\ u ,m}(z):=\\Gamma ^{m}(\\lambda +1) \\Gamma (\\lambda +\\mu +1)2^{2\\lambda +\\mu}z^{1-(\ u /2)-\\lambda} \\mathcal{J}_{\\lambda ,\\mu }^{\ u ,m}(\\sqrt{z})$\\end{document}, where the function Jλ,μν,m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{J}_{\\lambda ,\\mu}^{\ u ,m}$\\end{document} satisfies the differential equation Jλ,μν,m(z):=(1−2λ−ν)Jλ,μν,m(z)+z(Jλ,μν,m(z))′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{J}_{\\lambda ,\\mu}^{\ u ,m}(z):=(1-2\\lambda -\ u )J_{ \\lambda ,\\mu}^{\ u ,m}(z)+z (J_{\\lambda ,\\mu }^{\ u ,m}(z) )^{\\prime}$\\end{document} with Jν,λμ,m(z)=(z2)2λ+ν∑k=0∞(−1)kΓm(k+λ+1)Γ(kμ+ν+λ+1)(z2)2k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ J_{\ u ,\\lambda}^{\\mu ,m}(z)= \\biggl(\\frac{z}{2} \\biggr)^{2\\lambda + \ u} \\sum_{k=0}^{\\infty} \\frac{(-1)^{k}}{\\Gamma ^{m} (k+\\lambda +1 )\\Gamma (k\\mu +\ u +\\lambda +1 )} \\biggl(\\frac{z}{ 2} \\biggr)^{2k} $$\\end{document} for λ∈C∖Z−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\in \\mathbb{C}\\setminus \\mathbb{Z}^{-}$\\end{document}, Z−:={−1,−2,−3,…}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{Z}^{-}:= \\{ -1,-2,-3,\\ldots \\}$\\end{document}, m∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$m\\in \\mathbb{N}$\\end{document}, ν∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ u \\in \\mathbb{C}$\\end{document}, and μ∈N0:=N∪{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu \\in \\mathbb{N}_{0}:=\\mathbb{N}\\cup \\{0\\}$\\end{document}. In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, 0≤α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0\\leq \\alpha <1$\\end{document}. Ultimately, we discuss the starlikeness and convexity of order zero for Jλ,μν,m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{J}_{\\lambda ,\\mu} ^{\ u ,m}$\\end{document}, and it turns out that they are useful to extend the range of validity for the parameter λ to λ≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\geq 0$\\end{document} where the main concept of the proofs comes from some technical manipulations given by Mocanu (Libertas Math. 13:27–40, 1993). Our results improve, complement, and generalize some well-known (nonsharp) estimates.

Some properties of convex and increasing convex orders under Archimedean copula

Abstract In this paper, the ordering properties of convex and increasing convex orders of the dependent random variables are studied. Some closure properties of the convex and increasing convex orders under independent random variables are extended to the dependent random variables under the Archimedean copula. Two applications are provided to illustrate our results.

Geometric Properties of Normalized Galué Type Struve Function

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ, convexity of order δ, k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature.

Increasing convex order of capital allocation with dependent assets under threshold model

In this manuscript, we consider a risk-preference investor allocating some amount of capital to the dependent risky asset, where the responding asset will occur default if the stochastic return is less than some predetermined threshold. Then, we present sufficient conditions of the increasing convex order on capital allocation with dependent risky assets when the stochastic return is right tail weakly stochastic arrangement increasing. Finally, some numerical examples are given as illustrations.

Stochastic ordering of variability measure estimators

Variability measures, such as the variance or the Gini mean difference, are widely used to summarize the dispersion of random variables. In the statistical setting, it is quite natural to assume that if a random quantity has more variability, then the estimators of its variability measures should be greater in some stochastic sense. Stochastic orders can be used to give inequalities in this regard, confirming, or not, this suitable property. This paper is devoted to the stochastic comparison of some variability measure estimators; conditions such that some of these estimators are comparable in the usual stochastic order and in the increasing convex order whenever the involved random variables have different variability are provided. Special attention is devoted to the cases of sample variance and Gini mean differences, and to the case of simple random samples.

Adjusted higher-order expected shortfall

How to detect different tail behaviors of two risk random variables with the same mean is an important task. In this paper, motivated by Burzoni et al. (2022), a class of convex risk measures, referred to as adjusted higher-order Expected Shortfall (ES), is introduced and studied. The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its higher-order ES does not exceed a pre-specified threshold for every probability level. This new risk measure is intimately linked to dual higher-order increasing convex order by choosing the risk threshold to be the higher-order ES of a special benchmark random loss. The dual representation for (adjusted) higher-order Expected Shortfall is also given.

Convex contractions on extended $ b $-metric spaces

&lt;abstract&gt;&lt;p&gt;This research investigated different types of convex contractions in the setting of extended $ b $-metric spaces from the point of view of the existence and uniqueness of their fixed points. The assumptions imposed on involved mappings refer to convexity of order $ 2 $, two-sided convexity or Ćirić-type convexity, which also fulfill a continuity type condition. An example was provided to emphasize the usability of the results.&lt;/p&gt;&lt;/abstract&gt;

Geometric Properties and Hardy Spaces of Rabotnov Fractional Exponential Functions

The aim of this study is to investigate a certain sufficiency criterion for uniform convexity, strong starlikeness, and strong convexity of Rabtonov fractional exponential functions. We also study the starlikeness and convexity of order γ. Moreover, we find conditions so that the Rabotnov functions belong to the class of bounded analytic functions and Hardy spaces. Various consequences of these results are also presented.

Ordering Results of Aggregate Claim Amounts from Two Heterogeneous Portfolios

In this paper, we discuss the stochastic comparison of two classical surplus processes in an one-year insurance period. Under the Marshall-Olkin extended Weibull random aggregate claim amounts, we establish some sufficient conditions for the comparison of aggregate claim amounts in the sense of the usual stochastic order. Applications of our results to the Value-at-Risk and ruin probability are also given. The obtained results show that the heterogeneity of the risks in a given insurance portfolio tends to make the portfolio volatile, which in turn leads to requiring more capital. We also obtain some sufficient conditions for comparing aggregate non-random claim amounts with different occurrence frequency vectors in terms of increasing convex order.

Functional convex order for the scaled McKean–Vlasov processes

Functional convex order for the scaled McKean–Vlasov processes

Supermodular and directionally convex comparison results for general factor models

This paper provides comparison results for general factor models with respect to the supermodular and directionally convex order. These results extend and strengthen previous ordering results from the literature concerning certain classes of mixture models as mixtures of multivariate normals, multivariate elliptic and exchangeable models to general factor models. For the main results, we first strengthen some known orthant ordering results for the multivariate ▪ -product of the specifications, which represents the copula of the factor model, to the stronger notion of the supermodular ordering. The stronger comparison results are based on classical rearrangement results and in particular are rendered possible by some involved constructions of transfers as arising from mass transfer theory. The ordering results for ▪ -products are then extended to factor models with general conditional dependencies. As a consequence of the ordering results, we derive worst case scenarios in relevant classes of factor models allowing, in particular, interesting applications to deriving sharp bounds in financial and insurance risk models. The results and methods of this paper are a further indication of the particular effectiveness of Sklar‘s copula notion.

Quantum Dilogarithm Identities Arising from the Product Formula for the Universal R-Matrix of Quantum Affine Algebras

In Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1–25), four quantum dilogarithm identities containing infinitely many factors are proposed as wall-crossing formulas for the refined BPS invariant. We give an algebraic proof of these identities using the formula for the universal R-matrix of the quantum affine algebra developed by Ito (Hiroshima Math. J. 40 (2010), 133–183), which yields various product presentations of the universal R-matrix by choosing various convex orders on an affine root system. By the uniqueness of the universal R-matrix and appropriate degeneration, we can construct various quantum dilogarithm identities, including the ones proposed in Dimofte, Gukov, and Soibelman (Lett. Math. Phys. 95 (2011), 1–25), which turn out to correspond to convex orders of multiple row type.

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

An optimal transport-based characterization of convex order

Abstract For probability measures μ , ν \mu ,\nu , and ρ \rho , define the cost functionals C ( μ , ρ ) ≔ sup π ∈ Π ( μ , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) and C ( ν , ρ ) ≔ sup π ∈ Π ( ν , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) , C\left(\mu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\mu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y)\hspace{1.0em}{\rm{and}}\hspace{1em}C\left(\nu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\nu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y), where ⟨ ⋅ , ⋅ ⟩ \langle \cdot ,\cdot \rangle denotes the scalar product and Π ( ⋅ , ⋅ ) \Pi \left(\cdot ,\cdot ) is the set of couplings. We show that two probability measures μ \mu and ν \nu on R d {{\mathbb{R}}}^{d} with finite first moments are in convex order (i.e., μ ≼ c ν \mu {\preccurlyeq }_{c}\nu ) iff C ( μ , ρ ) ≤ C ( ν , ρ ) C\left(\mu ,\rho )\le C\left(\nu ,\rho ) holds for all probability measures ρ \rho on R d {{\mathbb{R}}}^{d} with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of ∫ f d ν − ∫ f d μ \int f{\rm{d}}\nu -\int f{\rm{d}}\mu over all 1-Lipschitz functions f f , which is obtained through optimal transport (OT) duality and the characterization result of OT (couplings) by Rüschendorf, by Rachev, and by Brenier. Building on this result, we derive new proofs of well known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.

Backward and forward Wasserstein projections in stochastic order

Motivated by applications of stochastic orders in statistics and economics, we study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Duality theorems for backward and forward projections are established under mild conditions, which lay a foundation for investigating sampling of measures in stochastic orders. To demonstrate applications of these theorems beyond practice, particular attention is given to mathematical properties of convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler, particularly the existence and regularity of optimal mappings. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. The factorization of convex order projection completes the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to earlier results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.