A geometric mean inequality and some monotonicity results for the q-Gamma function
نویسندگان
چکیده
منابع مشابه
Monotonicity Results for the Gamma Function
The function [Γ(x+1)] 1/x x+1 is strictly decreasing on [1,∞), the function [Γ(x+1)]1/x √ x is strictly increasing on [2,∞), and the function [Γ(x+1)] 1/x √ x+1 is strictly increasing on [1,∞), respectively. From these, some inequalities, for example, the Minc-Sathre inequality, are deduced, and two open problems posed by the second author are solved partially.
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Letting P (u, x) denote the regularised incomplete gamma function, it is shown that for each α ≥ 0, P (x, x+ α) decreases as x increases on the positive real semiaxis, and P (x, x + α) converges to 1/2 as x tends to infinity. The statistical significance of these results is explored.
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Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function θa,b(x) = [Γ(x + b)/Γ(x + a)]1/(b−a) − x is strictly convex and decreasing on (−a,∞) with θa,b(∞) = a+b−1 2 and θa,b(−a) = a, where Γ denotes the Euler’s gamma function.
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ژورنال
عنوان ژورنال: Mathematical Inequalities & Applications
سال: 1998
ISSN: 1331-4343
DOI: 10.7153/mia-01-24