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Even coefficient estimates for bounded univalent functions

100%

Prokhorov D.

Annales Polonici Mathematici

1993

tom 58

nr 3

267-273

Extremal coefficient properties of Pick functions are proved. Even coefficients of analytic univalent functions f with |f(z)| < M, |z| < 1, are bounded by the corresponding coefficients of the Pick functions for large M. This proves a conjecture of Jakubowski. Moreover, it is shown that the Pick functions are not extremal for a similar problem for odd coefficients.

HCM property and the half-Cauchy distribution

75%

Bosch P.

Probability and Mathematical Statistics

2015

tom Vol. 35, Fasc. 2

191--200

Let Zα and Zα be two independent positive α-stable random variables. It is known that (Zα/Zα)α is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation (Zα/Zα)β is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if α ≤ 1/2 and |β| ≥ α/(1−α). This clarifies a conjecture of Bondesson (1992) on positive stable densities.