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Inequalities for Means in Two Variables
Abstract. 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
A monotonicity property of the gamma function
In this paper we obtain a monotoneity property for the gamma function that yields sharp asymptotic estimates for 17(x) as x tends to oc, thus proving a conjecture about 17(x).
Some properties of the gamma and psi functions, with applications
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
Generalized elliptic integrals and modular equations
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
Optimal combinations bounds of root-square and arithmetic means for Toader mean
We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1S(a,b) + (1 − α1) A(a,b) < T(a,b) < β1S(a,b) + (1 − β1) A(a,b) and
Sharp estimates for complete elliptic integrals
Monotonicity and convexity properties of certain functions defined in terms of complete elliptic integrals are studied and sharp functional inequalities for these functions are obtained, thus
Infinite products and normalized quotients of ypergeometric functions
For $r \in (0,1)$ and $a \in (0,1)$ the authors consider the quotient of hypergeometric functions $$ \mu_a(r)\equiv c F(a,1-a;1;1-r^2)/F(a,1-a;1;r^2), $$ where the normalizing coefficient $c = \pi/(2