We introduce the notion of strongly h-convex functions (defined on a normed space) and present some properties and representations of such functions. We obtain a characterization of inner product spaces involving the notion of strongly h-convex functions. Finally, a Hermite–Hadamard–type inequality for strongly h-convex functions is given.

In Kirchheim-Magnani [7] the authors construct a left invariant distance ρ on the Heisenberg group such that the identity map id is 1-Lipschitz but it is not metrically differentiable anywhere. In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability. In fact we show that they construct something which fails shortly from being a dilatation...

2.1 Differentiability and first order conditions If a function f : R → R has partial derivatives ∂if(x) = limt→0 t−1(f(x + tei) − f(x)) which exist and are continuous on R, it is a C1(R) function, and is differentiable at every x in the sense that f(x + h) − f(x) −∇f(x) · h = o(‖h‖) as h → 0. This means it can be approximated linearly, and the derivative is the linear map on R given by Df(x)(h)...

We study differentiability properties in a particular case of the Palmer’s linearization Theorem, which states the existence of an homeomorphism H between the solutions of a linear ODE system having exponential dichotomy and a quasilinear system. Indeed, if the linear system is uniformly asymptotically stable, sufficient conditions ensuring that H is a C preserving orientation diffeomorphism ar...

Abstract Some new integral inequalities for strongly $(\alpha ,h-m)$ ( α , h − m ) -convex functions via generalized Riemann–Liouville fractional integrals are established. The outcomes of this paper provide ...

Abstract We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. also provide two applications: a Lusin-type approximation ${\mathbb {H}}$-rectifiable sets and coarea formula that completes program started in [18].