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15
Generalized elliptic integrals
, 2004
"... Jacobi’s elliptic integrals and elliptic functions arise naturally from the SchwarzChristoffel conformal transformation of the upper half plane onto a rectangle. In this paper we study generalized elliptic integrals which arise from the analogous mapping of the upper half plane onto a quadrilateral ..."
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Cited by 7 (5 self)
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Jacobi’s elliptic integrals and elliptic functions arise naturally from the SchwarzChristoffel conformal transformation of the upper half plane onto a rectangle. In this paper we study generalized elliptic integrals which arise from the analogous mapping of the upper half plane onto a quadrilateral and obtain sharp monotonicity and convexity properties for certain combinations of these integrals, thus generalizing analogous wellknown results for classical conformal capacity and quasiconformal distortion functions.
On l’Hospitaltype rules for monotonicity
 J. Inequal. Pure Appl. Math
"... ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio r: = f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ: = f ′ /g ′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for li ..."
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Cited by 7 (7 self)
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ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio r: = f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ: = f ′ /g ′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for limits, since in general we do not require that both f and g, or either of them, tend to 0 or ∞ at an endpoint or any other point of (a, b). Here new insight into the nature of the rules for monotonicity is provided by a key lemma, which implies that, if ρ is monotonic, then ˜ρ: = r ′ · g 2 /g ′  is so; hence, r ′ changes sign at most once. Based on the key lemma, a number of new rules are given. One of them is as follows: Suppose that f(a+) = g(a+) = 0; suppose also that ρ ↗ ↘ on (a, b) – that is, for some c ∈ (a, b), ρ ↗ (ρ is increasing) on (a, c) and ρ ↘ on (c, b). Then r ↗ or ↗ ↘ on (a, b). Various applications and illustrations are given.
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.
On conformal moduli of polygonal quadrilaterals. Helsinki preprint 417
, 2005
"... Abstract. The change of conformal moduli of polygonal quadrilaterals under some geometric transformations is studied. We consider the motion of one vertex when the other vertices remain fixed, the rotation of sides, polarization, symmetrization, and averaging transformation of the quadrilaterals. So ..."
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Cited by 4 (4 self)
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Abstract. The change of conformal moduli of polygonal quadrilaterals under some geometric transformations is studied. We consider the motion of one vertex when the other vertices remain fixed, the rotation of sides, polarization, symmetrization, and averaging transformation of the quadrilaterals. Some open problems are formulated.
Turán type inequalities for hypergeometric functions
 Proc. Amer. Math. Soc
"... Dedicated to the memory of Professor Alexandru Lupa¸s Abstract. In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some ..."
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Cited by 4 (1 self)
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Dedicated to the memory of Professor Alexandru Lupa¸s Abstract. In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems. 1.
L’Hospitaltype rules for monotonicity, and the Lambert and Saccheri quadrilaterals in hyperbolic geometry
 ONLINE: http:// jipam.vu.edu.au/article.php?sid=573
"... ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio f/g of two functions on an interval of the real line based on the monotonicity pattern of the ratio f ′ /g ′ of the derivatives. These rules are applicable even more broadly than the l’Hospital rules for limits, since w ..."
Abstract

Cited by 4 (4 self)
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ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio f/g of two functions on an interval of the real line based on the monotonicity pattern of the ratio f ′ /g ′ of the derivatives. These rules are applicable even more broadly than the l’Hospital rules for limits, since we do not require that both f and g, or either of them, tend to 0 or ∞ at an endpoint of the interval. Here these rules are used to obtain monotonicity patterns of the ratios of the pairwise distances between the vertices of the Lambert and Saccheri quadrilaterals in the Poincaré model of hyperbolic geometry. Some of the results may seem surprising. Apparently, the methods will work for other ratios of distances in hyperbolic geometry and other Riemann geometries. The presentation is mainly selfcontained.
SHARPENING OF JORDAN’S INEQUALITY AND ITS APPLICATIONS
, 2006
"... ABSTRACT. In this paper,the following inequality: 2 1 π 2π5 (π4 − 16x 4 sin x 2 π − 2 x π π5 (π 4 − 16x 4) is established. An application of this inequality gives an improvement of Yang Le’s inequality. ..."
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Cited by 4 (0 self)
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ABSTRACT. In this paper,the following inequality: 2 1 π 2π5 (π4 − 16x 4 sin x 2 π − 2 x π π5 (π 4 − 16x 4) is established. An application of this inequality gives an improvement of Yang Le’s inequality.
On rationally parametrized modular equations
"... Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformati ..."
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Cited by 3 (0 self)
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Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformation of elliptic curve periods, which are parametrized by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equation, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algebraic transformation formulas for special functions. The ones for N = 4,3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3,4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.
Moments of Ramanujan’s Generalized Elliptic Integrals and Extensions of Catalan’s Constant
, 2010
"... We undertake a thorough investigation of the moments of Ramanujan’s alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalanrelated constants and various new hypergeometric identities. 1 Introduction and ..."
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Cited by 1 (0 self)
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We undertake a thorough investigation of the moments of Ramanujan’s alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalanrelated constants and various new hypergeometric identities. 1 Introduction and
Generalized Elliptic Integrals II
, 2007
"... We study monotonicity and convexity properties of functions arising in the theory of elliptic integrals, and in particular in the case of a SchwartzChristoffel conformal mapping from a halfplane to a trapezoid. We obtain sharp monotonicity and convexity results for combinations of these functions, ..."
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We study monotonicity and convexity properties of functions arising in the theory of elliptic integrals, and in particular in the case of a SchwartzChristoffel conformal mapping from a halfplane to a trapezoid. We obtain sharp monotonicity and convexity results for combinations of these functions, as well as functional inequalities and a linearization property.