f " ' f " 2 where Sf := (p-) 1⁄2 (iv) denotes the Schwarzian derivative, and that this result is sharp, as the function f(z) = 1⁄2 log 11 z + z shows. Tile method of the proof does not give all sharp functions. We shall show that the sharp function is essentially unique using another approach implying Nehari's result. This shows furthermore that all convex domains except of parallel strip domai...

In this paper, we consider an integer convex optimization problem where the objective function is the sum of separable convex functions (that is, of the form Σ(i,j)∈Q ij ij F (w ) + Σi∈P i i B ( ) μ ), the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the form μi μj ≤ wij, (i, j) ∈ Q), with lower and upper bounds on variables. Let n = |P|, m = ...

Learning with non-modular losses is an important problem when sets of predictions are made simultaneously. The main tools for constructing convex surrogate loss functions for set prediction are margin rescaling and slack rescaling. In this work, we show that these strategies lead to tight convex surrogates iff the underlying loss function is increasing in the number of incorrect predictions. Ho...

The dynamic maintenance of the convex hull of a set of points in the plane is one of the most important problems in computational geometry. We present a data structure supporting point insertions in amortized O(log n · log log log n) time, point deletions in amortized O(log n · log log n) time, and various queries about the convex hull in optimal O(log n) worst-case time. The data structure req...

assume that $mathbb{d}$ is the open unit disk. applying ozaki's conditions, we consider two classes of locally univalent, which denote by $mathcal{g}(alpha)$ and $mathcal{f}(mu)$ as follows begin{equation*}  mathcal{g}(alpha):=left{fin mathcal{a}:mathfrak{re}left( 1+frac{zf^{prime prime }(z)}{f^{prime }(z)}right)

Borwein, Affleck, and Girgensohn (2000) posed a problem concerning the shape (that is, convexity, log-convexity, reciprocal concavity) of a certain function of several arguments that had manifested in a number of contexts concerned with optimization problems. In this paper we further explore the shape of the Borwein-Affleck-Girgensohn function and especially its extensions generated by complete...

We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ ℝ(d) and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y) = e(-y) for y ∈ ℝ; in this case, the resulting class of densities [Formula: see text]is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails tha...

We consider the problem of reporting convex hull points in an orthogonal range query in two dimensions. Formally, let P be a set of n points in R. A point lies on the convex hull of a point set S if it lies on the boundary of the minimum convex polygon formed by S. In this paper, we are interested in finding the points that lie on the boundary of the convex hull of the points in P that also fal...

In the paper, after reviewing the history, background, origin, and applications of the functions b t −a t t and e −αt −e −βt 1−e−t , we establish sufficient and necessary conditions such that the special function e αt −e βt eλt−eμt are monotonic, logarithmic convex, logarithmic concave, 3-log-convex and 3-log-concave on R, where α, β, λ and μ are real numbers satisfying (α, β) 6= (λ, μ), (α, β)...