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The authors present sharp Landen transformation inequalities for the hypergeometric functions F 1 2 ( a , b ; a + b ; x ) and F 1 2 ( a , b ; ( a + b + 1 ) / 2 ; x ) , by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear (or rational) functions, thus giving complete solutions of the problem on extending the well-known Landen transformation identities for the complete elliptic integrals of the first kind to these two hypergeometric functions, and substantially improving the related known results. As applications of these results, sharp Landen transformation inequalities are obtained for the generalized Grotzsch ring functions and the modular functions, which appear in Ramanujan's modular equations. Some other properties of hypergeometric functions and several properties of the Ramanujan constant are obtained, too.

The authors obtain some monotonicity and concavity properties, and asymptotically sharp bounds for the generalized elliptic integral K a ( r ) of the first kind and its special case K ( r ) , the complete elliptic integral of the first kind, by studying the monotonicity and concavity properties of certain combinations defined in terms of the Gaussian hypergeometric function F ( a , 1 − a ; 1 ; x ) and elementary functions, thus improving some well-known results.

In this paper, we study the quotient of hypergeometric functions μ a (r) in the theory of Ramanujan’s generalized modular equation for a ∈ (0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.

For a ∈ ( 0 , 1 / 2 ] and r ∈ ( 0 , 1 ) , let K a ( r ) and E a ( r ) ( K ( r ) and E ( r ) ) denote the generalized elliptic integrals (the complete elliptic integrals, respectively) of the first and second kinds, respectively. In this paper, the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a ∈ ( 0 , 1 / 2 ] . By these results, sharp bounds expressed in terms of K ( r ) , E ( r ) and elementary functions are obtained for K a ( r ) and E a ( r ) , thus showing the dependence on the parameter a of K a ( r ) and E a ( r ) , and substantially improving the known related results.

In this paper, we prove that the double inequality \begin{equation*} 1+\alpha r'^2, Comment: 14 pages

In this paper, we present some monotonicity theorems involving the generalized elliptic integrals Ka(r) and Ea(r) , and find an asymptotic property of Ka(r) when r → 1 . As applications, some well known results are improved. Mathematics subject classification (2010): 33E05.

In this paper, we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to Holder means.

In this paper, we present some sharp bounds for complete elliptic integrals of the second kind. As applications, we establish several inequalities for the perimeter of an ellipse.

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) 0 with a ≠ b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a,b) = [(a2 + b2)/2]1/2, A(a,b) = (a + b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.

In this paper, the authors prove some inequalities for the generalized elliptic integrals with respect to Holder means.

In this paper, we study the monotonicity properties of the generalized elliptic integrals and establish two Holder mean inequalities for the generalized Grotzsch ring function μa(r) and the generalized Hersch-Pfluger distortion function φa K(r) . Mathematics subject classification (2010): 30C62, 33E05.

We prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind, K and E. These results lead to sharp symmetrical bounds for K and E, which improve recently discovered inequalities.

In this paper the authors study a monotonicity of several functions involving m a ( r ) and μ a ( r ) . By using these results, the authors obtain some new bounds of the solutions of Ramanujan's generalized modular equations.

The authors prove a conjecture on elliptic integrals and obtain sharp bounds for φ K ( r ) and λ ( K ) . By using Teichmuller's module theorem, the authors obtain a distortion theorem of K-quasiconformal mappings on the plane.

In this work, we prove that M p ( K ( r ) , E ( r ) ) > π / 2 for all r ∈ ( 0 , 1 ) if and only if p ≥ − 1 / 2 , where M p ( x , y ) denotes the power mean of order p of two positive numbers x and y , and K ( r ) and E ( r ) denote the complete elliptic integrals of the first and second kinds, respectively.