Cauchy's means of Levinson type.
Companion inequalities to Jensen's inequality for -convex and -convex functions.
Complements of Lyapunov's Inequality.
Complete extension of a convex function
Continuity properties of convex-type set-valued maps.
Convex functions and the Hadamard inequality.
Convex functions of higher orders in Euclidean spaces
Convex functions with respect to logarithmic mean and ѕandwich theorem
Convex transformations with Banach lattice range.
A closed epigraph theorem for Jensen-convex mappings with values in Banach lattices with a strong unit is established. This allows one to reduce the examination of continuity of vector valued transformations to the case of convex real functionals. In particular, it is shown that a weakly continuous Jensen-convex mapping is continuous. A number of corollaries follow; among them, a characterization of continuous vector-valued convex transformations is given that answers a question raised by Ih-Ching...
Convexity and almost convexity in groups
We give a review of results proved and published mostly in recent years, concerning real-valued convex functions as well as almost convex functions defined on a (not necessarily convex) subset of a group. Analogues of such classical results as the theorems of Jensen, Bernstein-Doetsch, Blumberg-Sierpiński, Ostrowski, and Mehdi are presented. A version of the Hahn-Banach theorem with a convex control function is proved, too. We also study some questions specific for the group setting, for instance...
Convexity of the zero-balanced Gaussian hypergeometric functions with respect to Hölder means.
Convexity properties and inequalities for a generalized gamma function.
Convexity with given infinite weight sequences.
Convex-like inequality, homogeneity, subadditivity, and a characterization of -norm
Let a and b be fixed real numbers such that 0 < mina,b < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the -norm.
Critères de convexité et inégalités intégrales
Pour trois fonctions non-négatives intégrables sur , et , telles que , Borelll a établi l’inégalité . Nous déterminons les conditions précises où l’inégalité sera stricte. La clef de cette analyse est une nouvelle caractérisation des fonctions convexes mesurables.
Curves in Banach spaces which allow a parametrization or a parametrization with finite convexity
We give a complete characterization of those (where is a Banach space) which allow an equivalent parametrization (i.e., a parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for . We present examples which show applicability of our characterizations. For example, we show that the and parametrization problems are equivalent for but are not equivalent for .