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.

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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 well-known results for the volume Ωn of the unit ball B n ⊂ R n , the surface area ω n−1 of the unit sphere S n−1 , and some related constants.

We present various new inequalities involving the logarithmic mean L(x, y) = (x − y)/(log x − log y), the identric mean I (x, y) = (1/e)(x x /y y ) 1/(x−y) , and the classical arithmetic and geometric means, A(x, y) = (x + y)/2 and G(x, y) = √ xy. In particular, we prove the following conjecture, which was published in 1986 in this journal. If M r (x, y) = (x r /2+y r /2) 1/r (r = 0) denotes the power mean of order r, thenwith the best possible parameter c = (log 2)/(1 + log 2).