Logarithmic Convexity and Inequalities for the Gamma Function
نویسندگان
چکیده
منابع مشابه
Optimal inequalities for the power, harmonic and logarithmic means
For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...
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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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We prove the following two theorems: (i) Let Mr(a, b) be the rth power mean of a and b. The inequality Mr(Γ(x), Γ(1/x)) ≥ 1 holds for all x ∈ (0,∞) if and only if r ≥ 1/C − π2/(6C2), where C denotes Euler’s constant. This refines results established by W. Gautschi (1974) and the author (1997). (ii) The inequalities xα(x−1)−C < Γ(x) < xβ(x−1)−C (∗) are valid for all x ∈ (0, 1) if and only if α ≤...
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Write R(x, y) = Γ(x + y) Γ(x). Inequalities for this ratio have interesting applications, and have been considered by a number of writers over a long period. In a Monthly article [7], Wendel showed that x(x + y) y−1 ≤ R(x, y) ≤ x y for 0 ≤ y ≤ 1. (1) Wendel's method was an ingenious application of Hölder's inequality to the integral definition of the gamma function. Note that both inequalities ...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1996
ISSN: 0022-247X
DOI: 10.1006/jmaa.1996.0385