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 m 1 , m 2 ∈ M, we say that a function f :for all x, y ∈ R + . The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m 1 , m 2 )-convexity on m 1 and m 2 and give sufficient conditions for (m 1 , m 2 )-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.

Abstract. This paper considers the problem of defining Cauchy sequence and completeness in quasi-pseudo-metric spaces. The definitions proposed allow versions of such classical theorems as the Baire Category Theorem, the Contraction Principle and Cantor's characterization of completeness to be formulated in the quasi-pseudo-metric setting.

In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane 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.

Jacobi's elliptic integrals and elliptic functions arise naturally from the Schwarz-Christoffel 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 well-known results for classical conformal capacity and quasiconformal distortion functions. An algorithm for the computation of the modulus of the quadrilateral is given.2000 Mathematics Subject Classification: Primary 33B15, 33C05, Secondary 30C62.A generalized modular equation of order (or degree) p > 0 isSometimes we just call this an (a, b, c)-modular equation of order p and we usually assume that a, b, c > 0 with a + b ≥ c, in which case this equation uniquely defines s, see Lemma 4.5.Many particular cases of (1.2) have been studied in the literature on both analytic number theory and geometric function theory,The classical case (a, b, c) = ( 1 2 , 1 2 , 1) was studied already by Jacobi and many others in the nineteenth century, see [Be]. In 1995 B. Berndt, S. Bhargava, and F. Garvan published an important paper [BBG] in which they studied the case (a, b, c) = (a, 1 − a, 1) and p an integer. For several rational values of a such as a = 1 3 , 1 4 , 1 6 and integers p (e.g. p = 2, 3, 5, 7, 11, ...) they were able to give proofs for numerous algebraic identities stated by Ramanujan in his unpublished notebooks. These identities involve r and s from (1.2). After the publication of [BBG] many papers have appeared on modular equations, see e.g. [AQVV], [Be], [CLT], [Q], and [S].To rewrite (1.2) in a slightly shorter form, we use the decreasing homeomorphism µ a,b,c : (0, 1) → (0, ∞), defined bythe beta function, see (3.5) below. We call µ a,b,c the generalized modulus, cf. [LV, (2.2)]. We can now write (1.2) as µ a,b,c (s) = p µ a,b,c (r) , 0 < r < 1 . (1.4)With p = 1/K, K > 0, the solution of (1.2) is then given byWe call ϕ a,b,c K the (a, b, c)-modular function with degree p = 1/K [BBG], [AQVV, (1.5)]. In the case a < c we also use the notation µ a,c = µ a,c−a,c , ϕ a,c K = ϕ a,c−a,c K . For 0 < a < min{c, 1} and 0 < b < c ≤ a + b, define the generalized complete elliptic integrals of the first and second kinds (cf. [AQVV, (1.9), (1.10), (1.3), and (1.5)]) on [0, 1] by K = K a,b,c = K a,b,c (r) = B(a, b) 2 F (a, b; c; r 2 ) , (1.6) E = E a,b,c = E a,b,c (r) = B(a, b) 2 F (a − 1, b; c; r 2 ) ,

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