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A completely monotonic function involving divided differences of psi and polygamma functions and an application
 RGMIA Res. Rep. Coll
"... Abstract. A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw’s double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of t ..."
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Abstract. A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw’s double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw’s double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in R n−1 and R n respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized. 1.
Some logarithmically completely monotonic functions involving gamma function
, 2005
"... Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x ..."
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Cited by 11 (7 self)
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Abstract. In this article, logarithmically complete monotonicity properties of some functions such as 1 [Γ(x+1)] 1/x
Generalized Abstracted Mean Values
 J. Inequal. Pure and Appl. Math
, 2000
"... In this article, the author introduces the generalized abstracted mean values which extend the concepts of most means with two variables, and researches their basic properties and monotonicities. ..."
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Cited by 7 (4 self)
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In this article, the author introduces the generalized abstracted mean values which extend the concepts of most means with two variables, and researches their basic properties and monotonicities.
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 CONCAVITY PROPERTIES OF SOME FUNCTIONS INVOLVING THE GAMMA FUNCTION WITH APPLICATIONS
"... ABSTRACT. In this article, we give the monotonicity and concavity properties of some functions involving the gamma function and some equivalence sequences to the sequence n! with exact equivalence constants. ..."
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ABSTRACT. In this article, we give the monotonicity and concavity properties of some functions involving the gamma function and some equivalence sequences to the sequence n! with exact equivalence constants.
Beijing, People’s Republic of China
, 2010
"... We derive saddlepoint approximations for the distribution and density functions of the halflife estimated by OLS from autoregressive timeseries models. Our results are used to prove that none of the integerorder moments of these halflife estimators exist. This provides an explanation for the ver ..."
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We derive saddlepoint approximations for the distribution and density functions of the halflife estimated by OLS from autoregressive timeseries models. Our results are used to prove that none of the integerorder moments of these halflife estimators exist. This provides an explanation for the very large estimates of persistency, and the extremely wide confidence intervals, that have been reported by various authors – for example in the empirical economics literature relating to purchasing power parity.
Uniform bounds for the complementary incomplete gamma function, Preprint at http://locutus.cs.dal.ca:8088/archive/00000335
"... Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : z > ca − 1} with a real and z complex. Our results show that within the hyperboloid, ..."
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Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : z > ca − 1} with a real and z complex. Our results show that within the hyperboloid, Γ(a, z)  is of order z  a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.
(α, β), the standard Nuttall Q
, 712
"... (α, β), and the normalized Nuttall Qfunction, QM,N(α, β), with respect to their real order indices M, N. Besides, closedform expressions are derived for the computation of the standard and normalized Nuttall Qfunctions for the case when M, N are odd multiples of 0.5 and M ≥ N. By exploiting these ..."
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(α, β), and the normalized Nuttall Qfunction, QM,N(α, β), with respect to their real order indices M, N. Besides, closedform expressions are derived for the computation of the standard and normalized Nuttall Qfunctions for the case when M, N are odd multiples of 0.5 and M ≥ N. By exploiting these results, novel upper and lower bounds for Q M,N (α, β) and QM,N(α, β) are proposed. Furthermore, specific tight upper and lower bounds for Q M (α, β), previously reported in the literature, are extended for real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, in the informationtheoretic analysis of multipleinput multipleoutput systems and in the description of stochastic processes in probability theory, among others. Index Terms—Closedform expressions, generalized Marcum Qfunction, lower and upper bounds, monotonicity, normalized
On the Monotonicity of the Generalized Marcum and Nuttall
, 2009
"... Monotonicity criteria are established for the generalized Marcum Qfunction, QM(α, β), and the standard Nuttall Qfunction, QM,N(α, β). Specifically, we present that QM(α, β) is monotonically increasing with regard to its order M, for all ranges of the parameters α, β, whereas QM,N(α, β) possesses a ..."
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Monotonicity criteria are established for the generalized Marcum Qfunction, QM(α, β), and the standard Nuttall Qfunction, QM,N(α, β). Specifically, we present that QM(α, β) is monotonically increasing with regard to its order M, for all ranges of the parameters α, β, whereas QM,N(α, β) possesses analogous monotonicity behavior with respect to M + N, under the assumption of a ≥ 1. For the normalized Nuttall Qfunction, QM,N(α, β), we also state the same monotonicity criterion without the necessity of restricting the range of the parameter α. By exploiting these results, we propose closedform upper and lower bounds for the standard and normalized Nuttall Qfunctions. Furthermore, specific tight upper and lower bounds for QM(α, β), that have already been proposed in the literature for the case of integer M, are appropriately utilized in order to extend its validity over real values of M. The offered theoretical results can be efficiently applied in the study of digital communications over fading channels, the capacity analysis of multipleinput multipleoutput (MIMO) channels and the decoding of turbo or lowdensity paritycheck (LDPC) codes. Index Terms Digital communications over fading channels, generalized Marcum Qfunction, Nuttall Qfunction,
Some Analytical and Numerical Consequences of Sturm Theorems
, 2012
"... The Sturm comparison theorems for second order ODEs are classical results from which information on the properties of the zeros of special functions can be obtained. Sturm separation and comparison theorems are also available for differencedifferential systems under oscillatory conditions. The sepa ..."
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The Sturm comparison theorems for second order ODEs are classical results from which information on the properties of the zeros of special functions can be obtained. Sturm separation and comparison theorems are also available for differencedifferential systems under oscillatory conditions. The separation theorem provides interlacing information for zeros of some special functions and the comparison theorem gives bounds on the distance between these interlacing zeros. For monotonic systems Sturm theorems for the zeros do not exist because there is one zero at most. Instead, bounds on certain function ratios can be obtained using information on the coefficients of the system, and particularly monotonicity properties. Similar ideas that can be used to prove Sturm theorems can be considered for obtaining this type of bounds; the qualitative analysis of associated Riccati equations is a key ingredient in both cases. We review some applications for modified Bessel, parabolic cylinder and Laguerre functions and we also present related results for incomplete gamma functions. Sturm theorems, both for second order ODEs and first order DDEs can be applied for the computation of the real zeros of special functions. Recently a fourth order method based on the Sturm comparison theorem for computing the real zeros of solutions of second order ODEs was developed. We discuss the connection of this method with the Sturm theorem and we explain how this has been extended to the computation of complex zeros.