We determine the order of convexity of hypergeometric functions z 7→ F (a, b, c, z) as well as the order of starlikeness of shifted hypergeometric functions z 7→ zF (a, b, c, z), for certain ranges of the real parameters a, b and c. As a consequence we obtain the sharp lower bound for the order of convexity of the convolution (f ∗ g)(z) := ∑∞ n=0 anbnz n when f(z) = ∑∞ n=0 anz n is convex of or...

Problem of defining convexity of a digital region is considered. Definition of DL− (digital line) convexity is proposed, and it is shown to be stronger than the other two definitions, T−(triangle) convexity and L−(line) convexity. In attempt to connect the convexity of digital sets, to the fact that digital set can be a fuzzy set, the notion of convexity of the membership function is introduced...

We introduce and investigate the orbit-closed $C$-numerical range, a natural modification of range an operator introduced for $C$ trace-class by Dirr vom Ende. Our is conservative theirs because these two sets have same closure even coincide when finite rank. Since Ende's results concerning depend only on its closure, our inherits properties, but we also establish more. For selfadjoint, Ende we...

We study the convergence issue for gradient algorithm (employing general step sizes) optimization problems on Riemannian manifolds (without curvature constraints). Under assumption of local convexity/quasi-convexity (resp., weak sharp minima), local/global linear convergence) results are established. As an application, properties employing constant sizes and Armijo finding $L^p$ ($p\in[1,+\inft...

The practical effectiveness of Thm. A.1 is roughly commensurate with the difficulty of its proof. That is, the proof is essentially trivial, and the condition is nearly useless. There are very few problems where simple pointwise minimization of the integrand at every instant yields an arc that is even admissible, much less optimal! Even worse, we have met many true minimizers (often in problems...

We discuss domains of holomorphy and several notions of pseudoconvexity (drawing parallels with the corresponding concepts from geometric convexity), and present a mostly selfcontained solution to the Levi problem. We restrict our attention to domains of C. Full text

In this paper we extend the result of [6] on the characteristic of convexity of Orlicz spaces to the more general case of Musielak-Orlicz spaces over a non-atomic measure space. Namely, the characteristic of convexity of these spaces is computed whenever the Musielak-Orlicz functions are strictly convex.

This note investigates the convexity of the indefinite joint numerical range of a tuple of Hermitian matrices in the setting of Krein spaces. Its main result is a necessary and sufficient condition for convexity of this set. A new notion of “quasi-convexity” is introduced as a refinement of pseudo-convexity.

In the present paper an algorithm of C interpolation of discrete set of data is given using splines of arbitrary degree, which preserves the convexity of given set of data. Mathematics Subject Classification 2000: 65D05, 65D07, 41A05, 41A15.

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport...