There are many inequalities which in the class of continuous functions are equivalent to convexity (for example the Jensen inequality and the Hermite-Hadamard inequalities). We show that this is not a coincidence: every nontrivial linear inequality which is valid for all convex functions is valid only for convex functions.
Given a finite subset of , we study the continuous complex valued functions and the Schwartz complex valued distributions defined on with the property that the forward differences are (in distributional sense) continuous exponential polynomials for some natural numbers .
We give a characterization of the globally Lipschitzian composition operators acting in the space
We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.
The authors consider the difference equation where , , , , , and is a sequence of integers with and . They obtain results on the classification of the set of nonoscillatory solutions of () and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.
En choisissant des “caractères” et des “logarithmes”, méromorphes sur , construits à l’aide de la fonction Gamma d’Euler, et en utilisant des séries de factorielles convergentes, nous sommes en mesure, dans une première partie, de donner une “forme normale” pour les solutions d’un système aux différences singulier régulier. Nous pouvons alors définir une matrice de connexion d’un tel système. Nous étudions ensuite, suivant une idée de G.D. Birkhoff, le lien de celles-ci avec le problème de la classification...