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Inequalities for two sine polynomials

Horst Alzer, Stamatis Koumandos (2006)

Colloquium Mathematicae

We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have α ≤ ∑ j = 1 n - 1 1 / ( n ² - j ² ) s i n ( j x ) ≤ β , with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality 0 < ∑ j = 1 n - 1 ( n ² - j ² ) s i n ( j x ) holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

Józef Marcinkiewicz (1910-1940) - on the centenary of his birth

Lech Maligranda (2011)

Banach Center Publications

Józef Marcinkiewicz’s (1910-1940) name is not known by many people, except maybe a small group of mathematicians, although his influence on the analysis and probability theory of the twentieth century was enormous. This survey of his life and work is in honour of the 100 t h anniversary of his birth and 70 t h anniversary of his death. The discussion is divided into two periods of Marcinkiewicz’s life. First, 1910-1933, that is, from his birth to his graduation from the University of Stefan Batory in Vilnius,...

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