An analytic function f (z) in the unit disc D is called stable if sn(f,·)/f ≺ 1/f holds for all for n ∈ N0. Here sn stands for the nth partial sum of the Taylor expansion about the origin of f , and ≺ denotes the subordination of analytic functions in D. We prove that (1 − z)λ, λ ∈ [−1, 1], are stable. The stability of √(1 + z)/(1 − z) turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials

Fire spread inside a spacecraft is a constant concern in space travel. Understanding how the fire grows and spreads, and how it can potentially be extinguished is critical for planning future missions. The conditions inside a spacecraft can greatly vary from those encountered on earth, including microgravity, low velocity flows, reduced ambient pressure and high oxygen, and thus affecting the combustion processes. In microgravity, the contributions of thermal radiation from gaseous species and soot can play a critical role in the spread of a flame and the problem has not been fully understood yet. The overall objective of this work is to address this by studying the soot temperature of microgravity flames spreading over a thin solid in microgravity. The experiments presented here were performed as part of the NASA project Saffire IV, conducted in orbit on board the Cygnus resupply vehicle before it re-entered the Earth’s atmosphere. The fuel considered is a thin fabric made of cotton and fiberglass (Sibal) exposed to a forced flow of 20 cm/s in a concurrent flow configuration. Reconstruction of the flame temperature fields is extracted from two color broadband emission pyrometry (B2CP) as the flame propagates over the solid fuel. A methodology, relevant assumptions and its applicability to other microgravity experiments are discussed here. The data obtained shows that the technique provides an acceptable average temperature around K, which remains relatively constant during the spread with an error value smaller than 117 K. The data presented in this work provides a methodology that could be applied to other microgravity experiments to be performed by NASA. It is expected that the results will provide insight for what is to be expected in different conditions relevant for fire safety in future space facilities.