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35
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 26 (11 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 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 15 (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.
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 13 (11 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 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 11 (6 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.
Turán type inequalities for hypergeometric functions
 PROC. AMER. MATH. SOC
, 2008
"... 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. ..."
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Cited by 11 (2 self)
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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.
On conformal moduli of polygonal quadrilaterals
, 2008
"... 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 ..."
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Cited by 9 (5 self)
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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.
ON MODULI OF RINGS AND QUADRILATERALS: ALGORITHMS AND EXPERIMENTS
, 2009
"... Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of ..."
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Cited by 7 (3 self)
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Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new hpFEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the SchwarzChristoffel Toolbox of Driscoll and Trefethen.
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 6 (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.
L’HOSPITALTYPE RULES FOR MONOTONICITY, AND THE LAMBERT AND SACCHERI QUADRILATERALS IN HYPERBOLIC GEOMETRY
 JOURNAL OF INEQUALITIES IN PURE AND APPLIED MATHEMATICS
, 2005
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An identity related to Jordan’s inequality
 Article ID 76782
, 2006
"... The main purpose of this note is to establish an identity which states that the function sinx/x is a power series of (π2 − 4x2) with positive coefficients for all x = 0. This enable us to obtain a much stronger Jordan’s inequality than that obtained before. ..."
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Cited by 5 (1 self)
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The main purpose of this note is to establish an identity which states that the function sinx/x is a power series of (π2 − 4x2) with positive coefficients for all x = 0. This enable us to obtain a much stronger Jordan’s inequality than that obtained before.