The theme of these lectures is local and global properties of plurisubharmonic functions. First differential inequalities defining convex, subharmonic and plurisubharmonic functions are discussed. It is proved that the marginal function of a plurisubharmonic function is plurisubharmonic under certain hypotheses. We study the singularities of plurisubharmonic functions using methods from convexi...

In this paper, we introduce some new subclasses of analytic functions related to starlike, convex, close-to-convex and quasi-convex functions defined by using a generalized operator and the differential subordination principle. Inclusion relationships for these subclasses are established. Moreover, we introduce some integral-preserving properties. Key-Words: Starlike function; Convex function; ...

and Applied Analysis 3 Furthermore, if h is a convex function, then ∂ c h(?̂?) is in accordance with the subdifferential ∂h(?̂?) in convex analysis; that is,

It is well-known that the Dirichlet problem for the MongeAmpère equation det Du = μ in a bounded strictly convex domain Ω in R has a weak solution (in the sense of Aleksandrov) for any finite Borel measure μ on Ω and for any continuous boundary data. We consider the Dirichlet problem when Ω is only assumed to be convex, and give a necessary and sufficient condition on the boundary data for solv...

In this paper, we consider convex quadratic semidefinite optimization problems and provide a primal-dual Interior Point Method (IPM) based on a new kernel function with a trigonometric barrier term. Iteration complexity of the algorithm is analyzed using some easy to check and mild conditions. Although our proposed kernel function is neither a Self-Regular (SR) fun...

A restricted-orientation convex set, also called an O-convex set, is a set of points whose intersection with lines from some xed set is empty or connected. The notion of O-convexity generalizes standard convexity and orthogonal convexity. We explore some of the basic properties of O-convex sets in two and higher dimensions. We also study O-connected sets, which are restricted O-convex sets with...

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces. It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations. However, we will also show that the conv...

We study the minimization problem f (x) → min, x ∈ C, where f belongs to a complete metric space of convex functions and the set C is a countable intersection of a decreasing sequence of closed convex sets Ci in a reflexive Banach space. Let be the set of all f ∈ for which the solutions of the minimization problem over the set Ci converge strongly as i→∞ to the solution over the set C. In our r...

In this paper, we introduce general classes of generalized monotonicity and generalized convexity. For each type of (relatively) generalized monotone map, we establish a relationship to (relatively) generalized convex function. In this way, we obtain first-order characterizations for various (relatively) generalized convex functions. Our results extend/generalize similar result obtained in [3].

Constructive properties of uniform convexity, strict convexity, near convexity, and metric convexity in real normed linear spaces are considered. Examples show that certain classical theorems, such as the existence of points of osculation, are constructively invalid. The methods used are in accord with principles introduced by Errett Bishop.