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19
On some inequalities for the gamma and psi functions
 MATH. COMP
, 1997
"... We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, starshaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) = ..."
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Cited by 38 (1 self)
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We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, starshaped, and superadditive functions which are related to Γ and ψ. Euler’s gamma function Γ(x) =
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 16 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Generalized elliptic integrals and modular equations
 Pacific J. Math
"... In geometric function theory, generalized elliptic integrals and functions arise from the SchwarzChristoffel transformation of the upper halfplane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic num ..."
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Cited by 16 (8 self)
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In geometric function theory, generalized elliptic integrals and functions arise from the SchwarzChristoffel transformation of the upper halfplane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them. 1. Introduction. In 1995 B. Berndt, S. Bhargava, and F. Garvan published an important paper [BBG] in which they studied generalized modular equations and gave proofs for numerous statements concerning these equations made by Ramanujan in his unpublished notebooks. No record of Ramanujan’s original
Generalized convexity and inequalities
 The University of Auckland, Report Series
, 2006
"... Abstract. Let R+ = (0, ∞) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 ∈ M, we say that a function f: R+ → R+ is (m1, m2)convex if f(m1(x, y)) ≤ m2(f(x), f(y)) for all x, y ∈ R+. The usual convexity i ..."
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Cited by 5 (3 self)
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Abstract. Let R+ = (0, ∞) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 ∈ M, we say that a function f: R+ → R+ is (m1, m2)convex if f(m1(x, y)) ≤ m2(f(x), f(y)) for all x, y ∈ R+. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1, m2)convexity on m1 and m2 and give sufficient conditions for (m1, m2)convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function. 1.
A monotonicity property involving 3F2 and comparisons of classical approximations of elliptical arc length
 SIAM J. Math. Anal
"... Abstract. Conditions are determined under which 3F2 (−n, a, b; a + b +2,ε − n +1;1) is a monotone function of n satisfying ab · 3F2 (−n, a, b; a + b +2,ε − n +1;1)≥ab · 2F1 (a, b; a + b +2;1). Motivated by a conjecture of Vuorinen [Proceedings of Special Functions and Differential Equations, ..."
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Cited by 5 (3 self)
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Abstract. Conditions are determined under which 3F2 (−n, a, b; a + b +2,ε − n +1;1) is a monotone function of n satisfying ab · 3F2 (−n, a, b; a + b +2,ε − n +1;1)≥ab · 2F1 (a, b; a + b +2;1). Motivated by a conjecture of Vuorinen [Proceedings of Special Functions and Differential Equations,
SOME PROPERTIES OF THE GAMMA AND PSI FUNCTIONS, WITH APPLICATIONS
"... Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some wellknown results for the volume Ωn of the unit ball B n ⊂ R n,thesu ..."
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Cited by 5 (0 self)
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Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some wellknown results for the volume Ωn of the unit ball B n ⊂ R n,thesurface area ωn−1 of the unit sphere S n−1, and some related constants. 1.
GEOMETRIC PROPERTIES OF QUASICONFORMAL MAPS AND SPECIAL FUNCTIONS
, 2008
"... Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with several coauthors. In particular, we adopt result ..."
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Cited by 2 (1 self)
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Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with several coauthors. In particular, we adopt results from the book authored by AndersonVamanamurthyVuorinen [AVV6]. Part 1. Quasiconformal maps and spheres Part 2. Conformal invariants and special functions Part 3. Recent results on special functions
Refinements and sharpenings of some double inequalities for bounding the gamma function
 J. Inequal. Pure Appl. Math
"... ABSTRACT. In this paper, some sharp inequalities for bounding the gamma function Γ(x) and the ratio of two gamma functions are established. From these, several known results are recovered, refined, extended and generalized simply and elegantly. ..."
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Cited by 2 (2 self)
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ABSTRACT. In this paper, some sharp inequalities for bounding the gamma function Γ(x) and the ratio of two gamma functions are established. From these, several known results are recovered, refined, extended and generalized simply and elegantly.
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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Cited by 1 (1 self)
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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.
An approximation for zerobalanced Appell function F1 near (1, 1)
, 2007
"... Abstract. We suggest an approximation for the zerobalanced Appell hypergeometric function F1 near the singular point (1, 1). Our approximation can be viewed as a generalization of Ramanujan’s approximation for zerobalanced 2F1 and is expressed in terms of 3F2. We find an error bound and prove some ..."
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Abstract. We suggest an approximation for the zerobalanced Appell hypergeometric function F1 near the singular point (1, 1). Our approximation can be viewed as a generalization of Ramanujan’s approximation for zerobalanced 2F1 and is expressed in terms of 3F2. We find an error bound and prove some basic properties of the suggested approximation which reproduce the similar properties of the Appell function. Our approximation reduces to the approximation of CarlsonGustafson when the Appell function reduces to the first incomplete elliptic integral.