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NECESSARY AND SUFFICIENT CONDITIONS FOR A FUNCTION INVOLVING DIVIDED DIFFERENCES OF THE DI AND TRIGAMMA FUNCTIONS TO BE COMPLETELY MONOTONIC
, 903
"... Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two g ..."
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Cited by 7 (7 self)
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Abstract. In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions. 1.
A COMPLETE SOLUTION TO AN OPEN PROBLEM RELATING TO AN INEQUALITY FOR RATIOS OF GAMMA FUNCTIONS
, 902
"... Abstract. In this paper, we prove that for x + y> 0 and y + 1> 0 the inequality [Γ(x + y + 1)/Γ(y + 1)] 1/x s x + y [Γ(x + y + 2)/Γ(y + 1)] 1/(x+1) x + y + 1 is valid if x> 1 and reversed if x < 1, where Γ(x) is the Euler gamma function. This completely extends the result in [Y. Yu, An inequality fo ..."
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Cited by 2 (2 self)
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Abstract. In this paper, we prove that for x + y> 0 and y + 1> 0 the inequality [Γ(x + y + 1)/Γ(y + 1)] 1/x s x + y [Γ(x + y + 2)/Γ(y + 1)] 1/(x+1) x + y + 1 is valid if x> 1 and reversed if x < 1, where Γ(x) is the Euler gamma function. This completely extends the result in [Y. Yu, An inequality for ratios of gamma
(k − 1)! h (k−1)!
, 903
"... Abstract. The main aim of this paper is to prove that the double inequality ..."
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Abstract. The main aim of this paper is to prove that the double inequality
THE FUNCTION (b x − a x)/x: RATIO’S PROPERTIES
, 904
"... Abstract. In the paper, after reviewing the history, background, origin, and applications of the functions bt−a t and t e−αt−e −βt 1−e−t, we establish sufficient and necessary conditions such that the special function eαt−e βt eλt−e µt are monotonic, logarithmic convex, logarithmic concave, 3logco ..."
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Abstract. In the paper, after reviewing the history, background, origin, and applications of the functions bt−a t and t e−αt−e −βt 1−e−t, we establish sufficient and necessary conditions such that the special function eαt−e βt eλt−e µt are monotonic, logarithmic convex, logarithmic concave, 3logconvex and 3logconcave on R, where α, β, λ and µ are real numbers satisfying (α, β) = (λ, µ), (α, β) = (µ, λ), α = β and λ = µ. 1.