Logarithmically Complete Monotonicity Properties Relating to the Gamma Function
By Tie-hong Zhao, Y. Chu, Hua Wang. 2011
81 citations recorded by the source corpus.
monotonic on � 0, ∞� if � α, β� ∈{ � α, β� :1 / √ α ≤ β ≤ 1 ,α / 1 }∪{ � α, β� :0 <β ≤ 1 ,ϕ 1� α, β� ≥ 0 ,ϕ 2� α, β� ≥ 0} andfα,β� x�� −1 is strictly logarithmically completely monotonic on � 0, ∞� if � α, β� ∈ {� α, β� :0 <α ≤ 1/2,0 <β ≤ 1 }∪{ � α, β� :1 ≤ β ≤ 1/ √ α ≤ √ 2 ,α / 1 }∪{ � α, β� :1 /2 ≤ α< 1 ,β ≥ 1/� 1−α� } ,w hereϕ1� α, β ��� α 2 � α−1� β 2 �� 2α 2 −3α� 1� β−α and ϕ2� α, β ��� α−1� β 2 �� 2α 2 −5α� 2� β−1.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/deda580b986b10cf9c8876037274e8b22d3eacbd
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In Applied Mathematics
- Y. Chu
- Tie-hong Zhao
- Hong-Hu Chu
- Yu‐ming Chu
- Imran Abbas Baloch
- Miao-Kun Wang
- Wei-Mao Qian
- Saima Rashid
Citations in the corpus
- Some New Hermite–Hadamard-Type Inequalities Associated with Conformable Fractional Integrals and Their Applications (2020, unknown): “If α � 1, then we clearly see that inequalities (12) and (13) reduce to inequality (2).”
- Inequalities for generalized trigonometric and hyperbolic functions with one parameter (2020, unknown): “Actually, for p > 1, set arcsin ∗ p ( x ) = arcsin pp − 1 , p ( x ) = where B ( u , v ) = t u − 1 ( 1 − t ) v − 1 dt = Γ ( u ) Γ ( v ) / Γ ( u + v ) ( u , v > 0 ) is the classical Beta function [31, 37], and Γ ( u ) = + ∞ 0 t u − 1 e − t dt ( u > 0 ) is the Gamma function [79, 84].”
- Generation of new fractional inequalities via n polynomials s-type convexity with applications (2020, unknown): “where Γ (x) = ∫ ∞ 0 t x–1e–tdt is the Euler gamma function [72, 73].”
- Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means (2020, background): “0 xu−1e−xdx is the Euler gamma function [15, 53, 56].”
- Sharp Bounds for the Weighted Hölder Mean of the Zero-Balanced Generalized Complete Elliptic Integrals (2020, unknown): “(a + n − 1) = (a + n) (a) for n ∈ N, (x) = ∫ ∞ 0 t x−1e−t dt is the classical Euler gamma function [13,15,28, 33,35].”