By a given family of convex functions on the real axis that grow at infinity faster than any linear function and by certain logarithmically sequence positive numbers, we construct space infinitely differentiable line. Under condition logarithmic gap between weight functions, prove possibility approximation polynomials in this space.

We show that the L integral mean on rD of an analytic function in the unit disk D with respect to the weighted area measure (1 − |z|) dA(z), where −3 ≤ α ≤ 0, is a logarithmically convex function of r on (0, 1). We also show that the range [−3, 0] for α is best possible.

For 0 < p < ∞ and −2 ≤ α ≤ 0 we show that the L integral mean on rD of an analytic function in the unit disk D with respect to the weighted area measure (1−|z|) dA(z) is a logarithmically convex function of r on (0, 1).

Copyright q 2010 Y.-M. Chu and X.-M. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove that G x, y | ∫x y f t dt| is multiplicatively concave on a, b × a, b if f : a, b ⊂ 0,∞ → 0,∞ is continuous and multiplicatively ...

The present paper shows that any sequence of integers laying along a finite ray in any Pascal-pyramid is log-concave, consequently unimodal. Further, we describe an algorithm which provides the plateaus of the diagonal sequences locating on layer n of the regular Pascal pyramid.

Let S be a finite set with some rank function r such that the Whitney numbers w i = I { x e S I r ( x ) = i}l are log-concave. Given k, me N so that Wk-1 < Wk <~ Wk+m, set W = w k + w k + 1 + "'" + Wk + ~" Generalizing a theorem of Kleitman and Milner, we prove that every F ~ S with cardinality IFI/> Whas average rank at least ( k w k + ... + (k + m)wk+m)/W, provided the normalized profile vect...

In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in...