A remark on the coefficients of bounded polynomials

Let be such that |p(e^iq)|≤1 for ϕ∈R and |p(1)|=a∈[0,1]. An inequality of Dewan and Govil for the sum |a_v|+|a_n|, 0≤u<v≤n is sharpened.     

High-pressure phase studies on fluid binary mixtures of hydrocarbons with tetrafluoromethane and trifluoromethane between 273 and 630 K and up to 250 MPa.

Binary mixtures of hydrocarbons differing in molecular mass and/or structure with tetrafluoromethane (CF₄) or trifluoromethane (CHF₃) were investigated at pressures up to 250 MPa using the synthetic method. Measurements were performed in an optical high-pressure high-temperature autoclave. The experimental p(T) isopleths and the resulting critical curves and isothermal p(x) sections show that the systems CF₄ + heptane, CF₄ + dodecane, CF₄ + cis-decahydronaphthalene (cis-decalin), CF₄ + 1,2,3,4-tetrahydronaphthalene (tetralin), and CF₄ + 2,6,10,15,19,23-hexamethyltetracosane (squalane), as well as CHF₃ + octane, CHF₃ + decane, CHF₃ + cis-decalin, and CHF₃ + tetralin, belong to the so-called gas—gas equilibrium type of the second kind, whereas for CHF₃ + hexane the critical curve that corresponds to upper critical solution temperatures obviously ends on the three-phase surface liquid—liquid—gas.The results are discussed from a phase-theoretical point of view. The effect of the strong dipole CHF₃ (μ = 1.65 D) on the critical phase behaviour is compared with the critical behaviour observed in systems containing the non-polar compound CF₄. The critical lines of the systems studied exhibit a whole pattern of different shapes if one hydrocarbon component is replaced by another. Within the (CHF₃ + n-alkane) series the change from liquid—liquid to gas—gas critical behaviour is found from CHF₃ + hexane to CHF₃ + heptane.     

Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral means is important in certain problems in fluid dynamics, especially when the functions are non‐vanishing in the punctured unit disk . We consider the problem of finding the extremal function f which maximizes the integral means for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional where denotes the area of the image of under . The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.     

Extreme Bloch Functions and Summation Methods for Bloch Functions

In this paper we construct Bloch functions F for which the set {z | e sup _{|ζ| < 1} |F'(ζ)| ( 1 - |ζ| ² ) = |F'(z)| ( 1 - |z| ² )} is an analytic Jordan curve tangential to the unit disk in some points. It is proved that, using such functions, we can derive analogs to the Taylor expansion for Bloch functions in cases where the Taylor expansion does not converge. October 15, 1997. Date revised: March 12, 1998. Date accepted: June 18, 1998.