Abstract
In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of finitely many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.
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The authors are thankful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper.
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In memory of the first author’s mother, Ji-Rong Zhang, who passed away in December 1995.
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Qi, F., Li, WH., Yu, SB. et al. A ratio of finitely many gamma functions and its properties with applications. RACSAM 115, 39 (2021). https://doi.org/10.1007/s13398-020-00988-z
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DOI: https://doi.org/10.1007/s13398-020-00988-z
Keywords
- Ratio
- Gamma function
- Bernstein function
- Completely monotonic function
- Logarithmically completely monotonic function
- Inequality
- Multinomial coefficient
- Multivariate beta function
- Logarithmic derivative
- Logarithmic convexity
- Integral representation
- Open problem