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16
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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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.
Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, submitted
"... Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, clai ..."
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Abstract. In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel’s, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, LazarevićLupa¸s’s, Kershaw’s and ElezovićGiordanoPečarić’s inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years. Contents
Some properties of the psi and polygamma functions, Available online at http://arxiv.org/abs/0903.1003
"... Abstract. In this paper, some monotonicity and concavity results of several functions involving the psi and polygamma functions are proved, and then some known inequalities are extended and generalized. 1. ..."
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Abstract. In this paper, some monotonicity and concavity results of several functions involving the psi and polygamma functions are proved, and then some known inequalities are extended and generalized. 1.
Sharp inequalities for the psi function and harmonic numbers, Available online at
 http://arxiv.org/abs/0902.2524. INEQUALITIES FOR POLYGAMMA FUNCTIONS 11
"... Abstract. In this paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers Hn are established respectively, some results in [I. Muqattash and M. Yahdi, Infinite family of approximations of the Digamma function, Math. Comput. Modelling 43 (2006), 1329–1336.] are improved ..."
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Abstract. In this paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers Hn are established respectively, some results in [I. Muqattash and M. Yahdi, Infinite family of approximations of the Digamma function, Math. Comput. Modelling 43 (2006), 1329–1336.] are improved, and some remarks are given. 1.
AN ALTERNATIVE PROOF OF ELEZOVIĆGIORDANOPEČARIĆ’S THEOREM
, 903
"... Abstract. In the present note, an alternative proof is supplied for Theorem 1 ..."
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Abstract. In the present note, an alternative proof is supplied for Theorem 1
Complete monotonicity of some functions involving polygamma functions, submitted
"... Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) ..."
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Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) (x) are polygamma functions. 1.
Geometric convexity of a function involving gamma function and applications to inequality theory
 17; Available online at http://jipam.vu.edu.au/article.php?sid=830. (F. Qi) Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China Email address: qifeng618@gmail.com, qifeng618@hotm
"... ABSTRACT. In this paper, the geometric convexity of a function involving gamma function is studied, as applications to inequality theory, some important inequalities which improve some known inequalities, including Wallis ’ inequality, are obtained. ..."
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Cited by 3 (1 self)
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ABSTRACT. In this paper, the geometric convexity of a function involving gamma function is studied, as applications to inequality theory, some important inequalities which improve some known inequalities, including Wallis ’ inequality, are obtained.
Monotonicity and logarithmic convexity relating to the volume of the unit ball, submitted
"... Abstract. Let Ωn stand for the volume of the unit ball in Rn for n ∈ N. In the 1/(n ln n) present paper, we prove that the sequence Ωn is logarithmically convex 1/(n ln n) Ω and that the sequence is strictly decreasing for n ≥ 2. In n Ω 1/[(n+1)ln(n+1)] n+1 addition, some monotonic and concave prope ..."
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Abstract. Let Ωn stand for the volume of the unit ball in Rn for n ∈ N. In the 1/(n ln n) present paper, we prove that the sequence Ωn is logarithmically convex 1/(n ln n) Ω and that the sequence is strictly decreasing for n ≥ 2. In n Ω 1/[(n+1)ln(n+1)] n+1 addition, some monotonic and concave properties of several functions relating to Ωn are extended and generalized.
REFINEMENTS OF LOWER BOUNDS FOR POLYGAMMA FUNCTIONS
, 903
"... Abstract. In the paper, some lower bounds for polygamma functions are refined. 1. Introduction and ..."
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Abstract. In the paper, some lower bounds for polygamma functions are refined. 1. Introduction and
Supplements to a class of logarithmically completely monotonic functions associated with the gamma function
 Appl. Math. Comput
"... Abstract. In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on (0, ∞). As applications of the necessary and sufficient condition, some inequalities for bounding the ps ..."
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Abstract. In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on (0, ∞). As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived. This is a continuator of the paper [12]. 1.