From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions
نویسندگان
چکیده
منابع مشابه
Complete monotonicity involving some ratios of gamma functions
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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Ratios involving incomplete gamma functions and their monotonicity properties play important roles in financial risk analysis. We derive desired monotonicity properties either using Pinelis’ Calculus Rules or applying probabilistic techniques. As a consequence, we obtain several inequalities involving conditional expectations that have been of interest in actuarial science.
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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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In the article, a notion “logarithmically absolutely monotonic function” is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function ` 1+ α x ́x+β are proved, where α and β are given real parameters, a new proof for the inclusion t...
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Abstract In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2021
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2020.124478