Complete monotonicity properties of functions involving q-gamma and q-digamma functions
نویسندگان
چکیده
منابع مشابه
Complete Monotonicities of Functions Involving the Gamma and Digamma Functions
In the article, the completely monotonic results of the functions [Γ(x+ 1)]1/x, [Γ(x+α+1)]1/(x+α) [Γ(x+1)]1/x , [Γ(x+1)]1/x (x+1)α and [Γ(x+1)]1/x xα in x ∈ (−1,∞) for α ∈ R are obtained. In the final, three open problems are posed.
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We prove some properties of completely monotonic functions and apply them to obtain results on gamma and q-gamma functions.
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We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and q-gamma functions.
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In this paper, by using the properties of an auxiliary function, we mainly present the necessary and sufficient conditions for various ratios constructed by gamma functions to be respectively completely and logarithmically completely monotonic. As consequences, these not only unify and improve certain known results including Qi's and Ismail's conclusions, but also can generate some new inequali...
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Let L(x) := x − Γ(x+t) Γ(x+s) xs−t+1, where Γ(x) is Euler’s gamma function. We determine conditions for the numbers s, t so that the function Φ(x) := − Γ(x+t) xt−s−1 L′′(x) is strongly completely monotonic on (0, ∞). Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give se...
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ژورنال
عنوان ژورنال: Mathematical Inequalities & Applications
سال: 2014
ISSN: 1331-4343
DOI: 10.7153/mia-17-58