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The Complete Monotonicity of a Function Studied by Miller and Moskowitz

Horst Alzer — 2009

Bollettino dell'Unione Matematica Italiana

Let S ( x ) = l o g ( 1 + x ) + ∫ 0 1 [ 1 - ( 1 + t 2 ) x ] d ⁢ t log ⁡ t   and   F ( x ) = log 2 - S ( x ) ( 0 < x ∈ ℝ ) . We prove that F is completely monotonic on ( 0 , ∞ ) . This complements a result of Miller and Moskowitz (2006), who proved that F is positive and strictly decreasing on ( 0 , ∞ ) . The sequence { S ⁢ ( k ) } ( k = 1 , 2 , … ) plays a role in information theory.

On Ozeki’s inequality for power sums

Horst Alzer — 2000

Czechoslovak Mathematical Journal

Let p ∈ ( 0 , 1 ) be a real number and let n ≥ 2 be an even integer. We determine the largest value c n ( p ) such that the inequality ∑ i = 1 n | a i | p ≥ c n ( p ) holds for all real numbers a 1 , ... , a n which are pairwise distinct and satisfy min i ≠ j | a i - a j | = 1 . Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value c n ( p ) in the case p > 0 and n odd, and in the case p ≥ 1 and n even.

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