A Theory of Commodification

This chapter presents a novel theory of commodification. According to Marx, commodification is a process by which exchange value comes to dominate use value. Markets, money, and profit-orientation are instruments that facilitate the subjugation of use value to exchange value. However, the book takes market value, rather than exchange value, as the denominator of use value. Market value depends on supply and demand and is open to manipulation and speculation. The chapter, furthermore, argues that commodification is a process and, subsequently, distinguishes between formal, real, and fictitious commodification. Typically, real commodification (the transformation of goods and services) follows formal commodification (the imposition of a price), but in some cases products first have to be standardized in order to become commodities that can be exchanged on a market. Fictitious commodification, i.e., the introduction of quasi-markets in public services and the transformation of service users into consumers, prepares the ground for formal and real commodification.

Third Hankel determinants for two classes of analytic functions with real coefficients

In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

Estimates of Coefficient Functionals for Functions Convex in the Imaginary-Axis Direction

Let C0(h) be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula Re{(1−z2)f′(z)}>0. In this paper, some coefficient problems for C0(h) are considered. Some properties and bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates of the difference and of sum of successive coefficients, bounds of the sum of the first n coefficients and bounds of the n-th coefficient. The obtained results are used to determine coefficient estimates for both functions convex in the imaginary-axis direction with real coefficients and typically real functions. Moreover, the sum of the first initial coefficients for functions with a positive real part and with a fixed second coefficient is estimated.

Coefficient inequalities related with typically real functions

In this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

Koebe domains for the class of typically real functions that are convex in two directions

In this paper, we discuss the class of typically real functions that are convex in two directions. We determine the Koebe domain for this class and its different representations.

TOEPLITZ DETERMINANTS WHOSE ELEMENTS ARE THE COEFFICIENTS OF ANALYTIC AND UNIVALENT FUNCTIONS

Let ${\mathcal{S}}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ which are of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in ${\mathcal{S}}$ and certain of its subclasses. We also discuss similar problems for typically real functions.