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A characterization of the best L sub 1-approximation to a continuous function by classes of fixed knot polynomial splines which satisfy generalized convexity constraints is presented and uniqueness is shown. Included is the possibility of specifying the positivity, monotonicity, or convexity of the class. The proof of uniqueness uses recently developed results for Hermite-Birkhoff interpolation by splines. (Author)

In this paper we study a class of nonlinearities for which a nonlocal parabolic equation with Neumann-Robin boundary conditions, for $p$-Laplacian, has finite time blow-up solutions.

The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.

By making use of the familiar Mathieu series and its generalizations, the authors derive a number of new integral representations and present a systematic study of probability density functions and probability distributions associated with some generalizations of the Mathieu series. In particular, the mathematical expectation, variance and the characteristic functions, related to the probability density functions of the considered probability distributions are derived. As a consequence, some interesting inequalities involving complete monotonicity and log-convexity are derived.

In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman \cite{Lin}. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact $H$-twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, Meinrenken, Tolman, and Woodward \cite{Ler2} we extend our results to the case of Hamiltonian actions of general compact Lie groups on $H$-twisted generalized complex orbifolds.

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci curvature and the convexity of another weight function used in the definition of the generalized relative entropies. This convexity condition corresponds to Lott and Villani's version of the curvature-dimension condition. As applications, we obtain appropriate variants of the Talagrand, HWI and logarithmic Sobolev inequalities, as well as the concentration of measures. We also investigate the gradient flow of our generalized relative entropy.

We study generalized moment maps for a Hamiltonian action on a connected compact $H$-twisted generalized complex manifold introduced by Lin and Tolman and prove the convexity and connectedness properties of the generalized moment maps for a Hamiltonian torus action.

In the paper, the authors prove that the generalized sine function $\sin_{p,q}(x)$ and the generalized hyperbolic sine function $\sinh_{p,q}(x)$ are geometrically concave and geometrically convex, respectively. Consequently, the authors verify a conjecture posed in the paper "B. A. Bhayo and M. Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory 164 (2012), no.~10, 1415\nobreakdash--1426; Available online at \url{http://dx.doi.org/10.1016/j.jat.2012.06.003}".

Let R+ = (0,infinity) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 in M, we say that a function f : R+ to R+ is (m1,m2)-convex if f(m1(x,y)) < or = m2(f(x),f(y)) for all x, y in R+ . The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1,m2)-convexity on m1 and m2 and give sufficient conditions for (m1,m2)-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.

We investigate the Moreau-Yosida regularization and the associated proximal map in the context of discrete gradient flow for the 2-Wasserstein metric. Our main results are a stepwise contraction property for the proximal map and an "above the tangent line" inequality for the regularization. Using the latter, we prove a Talagrand inequality and an HWI inequality for the regularization, under appropriate hypotheses. In the final section, the results are applied to study the discrete gradient flow for R\'enyi entropies. As Otto showed, the gradient flow for these entropies in the 2-Wasserstein metric is a porous medium flow or a fast diffusion flow, depending on the exponent of the entropy. We show that a striking number of the remarkable features of the porous medium and fast diffusion flows are present in the discrete gradient flow and do not simply emerge in the limit as the time-step goes to zero.

Abstract Optimization under generalized convexity has emerged as a pivotal area in mathematical programming, extending classical convex analysis to encompass a broader spectrum of problems. This paper synthesizes existing secondary data to explore recent advancements and contributions in the field of optimization under generalized convexity. It delves into various generalized convexity concepts, including pseudo-convexity, quasi-convexity, and invex functions, and examines their applications in solving complex optimization problems. The literature review highlights key theoretical developments, algorithmic innovations, and practical applications across diverse domains such as economics, engineering, and machine learning. Additionally, the paper discusses the challenges and future prospects of optimization under generalized convexity, emphasizing the need for further research to enhance solution methodologies and extend applicability. By providing a comprehensive overview of the current state of the field, this study underscores the significance of generalized convexity in advancing optimization theory and practice.

Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized analytic functions $f$ defined in the domain $|z|

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a non-rigorous argument indicating that they are not displacement semiconvex.

How the dispersiveness of the mixing distribution carries over to the mixed model is qualified in terms of generalized convex functions. These ideas are extensions of those in Shaked (1980) and Schweder (1982). A representation akin to the one for dilations is also given for balayages defined in terms of these generalized convex functions. (Author)

We study the power mean inequality of the generalized trigonometric and hyperbolic functions with two parameters. The generalized $p$-trigonometric and $(p, q)$-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi, respectively.

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the $m$-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.

This paper considers generalized convexity and concavity properties of the optimal value function f* for the general parametric optimization problem P(e) of the form min x sub f (x,e) s.t. x epsilon R(e). Many results on convexity and concavity characterizations of f* were presented by the authors in a previous paper. Such properties of f* and the solution map S* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. The authors give sufficient conditions for several types of generalized convexity and concavity of f*, in terms of respective generalized convexity and concavity assumptions on f and convexity and concavity assumptions on the feasible region point-to-set map R. Specializations of these results to the general parametric inequality-equality constrained nonlinear programming problem and its right-hand-side version are provided. (Author)

We consider covariance estimation in the multivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is non-convex, yet it has been proved that its global solution can be often computed via simple fixed point iterations. Our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions. Our second contribution is a generalized framework for structured covariance estimation under sparsity constraints. We show that the optimizations can be formulated as convex minimization as long the MGGD shape parameter is larger than half and the sparsity pattern is chordal. These include, for example, maximum likelihood estimation of banded inverse covariances in multivariate Laplace distributions, which are associated with time varying autoregressive processes.

We study the convexity properties of the generalized trigonometric functions considered as functions of parameter. We show that $p\to\sin_p(y)$ and $p\to\cos_p(y)$ are log-concave on the appropriate intervals while $p\to\tan_p(y)$ is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle the major part of a conjecture put forward in a recent paper by Baricz, Bhayo and Vuorinen.

We introduce a new geometric characteristic of compact sets on the plane called $r$-convexity, which fits nicely into the concept of generalized convexity and extends essentially the conventional convexity. For a class of subharmonic functions on unbounded domains with $r$-convex compact complement, with the growth governed by the distance to the boundary, we obtain the Blaschke--type condition for their Riesz measures. The result is applied to the study of the convergence of the discrete spectrum for the Schatten--von Neumann perturbations of bounded linear operators in the Hilbert space.

In this paper, authors study the convexity and concavity properties of real-valued function with respect to the classical means, and prove a conjecture posed by Bruce Ebanks in \cite{e}.