Abstract. Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does. We discuss some Jackson type estimates where the constants involved depend on the location of the points of change of c...

We propose a notion of competitive equilibrium in an abstract setting called a Convex Economy using a concept of convexity borrowed from Convex Geometry. The "magic” of linear equilibrium prices is put into perspective in this abstract setting. The abstract notion of competitive equilibrium is applied to a variety of convex economies and versions of the first and second fundamental welfare theo...

This is a brief overview of some applications of the ideas of abstract convexity to the upper semilattices of gauges in finite dimensions.

Abstract. We prove that rank-(n−1) convexity does not imply S-quasiconvexity (i.e., quasiconvexity with respect to divergence free fields) in M for m > n, by adapting the well-known Šverák’s counterexample [5] to the solenoidal setting. On the other hand, we also remark that rank-(n − 1) convexity and Squasiconvexity turn out to be equivalent in the space of n×n diagonal matrices. This follows ...

Abstract We introduce the subclass Tj(n, m, γ, α, λ) of analytic functions with negative coefficients defined by generalized Sălăgean operator D λ . Coefficient estimates, some important properties of the class Tj(n, m, γ, α, λ) and distortion theorems are determined. Further, extremal properties and radii of close-to-convexity, starlikeness and convexity of the class Tj(n, m, γ, α, λ) are obta...

Abstract. We give an extension of Hua’s inequality in pre-Hilbert C∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is eq...

* Correspondence: [email protected] Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan Full list of author information is available at the end of the article Abstract We consider functionals due to the difference in Petrović and related inequalities and prove the log-convexity and exponential convexity of these functionals by using different families of functions...

G. Murugusundaramoorthy, S. Sivasubramanian, R. K. Raina Abstract. The authors introduce a new subclass UH(α, β, γ, δ, λ, k) of functions which are analytic in the open disc ∆ = {z ∈ C : |z| < 1}. Various results studied include the coefficient estimates and distortion bounds, radii of close-to-convexity, starlikeness and convexity and integral means inequalities for functions belonging to the ...

Abstract. The object of this paper is to show some properties of functions belonging to a subclass Mp(A, B, b, n) (where b is a complex number with Re(b) > 0 and A and B are two arbitrary constants with −1 6 B < A 6 1). Coefficient estimates and some distortion theorems for this class of functions are found. Radii of close-to-convexity, starlikeness and convexity are derived. An application to ...

Introducing certain types of morphisms for general (abstract) convexity spaces, we give several ways for reducing the (Calder-) Eckhoff partition problem to simpler equivalent forms (finite, point-convex, interval spaces; restricted form, i.e. with distinct points). With additional results (to appear in a forthcoming paper) we show how the general problem can be reduced to its restricted versio...