In the article, we establish a sharp double inequality involving the ratio of generalized complete elliptic integrals of the first kind, which is the improvement and generalization of some previously known results.

A sharp double inequality involving generalized complete elliptic integral of the first kind

In the article, we present an elegant double inequality for the ratio of the zero-balanced hypergeometric functions, which improve and refine some previously known results and also give a positive answer the question by proposed by Ismail.

On some refinements for inequalities involving zero-balanced hypergeometric function

In the article, we present the best possible bounds for the weighted Holder mean of the zero-balanced generalized complete elliptic integrals of the first and second kinds, which are the generalizations of previous results for complete p-elliptic integrals.

Sharp Bounds for the Weighted Hölder Mean of the Zero-Balanced Generalized Complete Elliptic Integrals

W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations

Based on a comparison principle, we discuss the existence, uniqueness and asymptotic behaviour of various boundary blow-up solutions, for a class of quasilinear elliptic equations, which are then used to obtain a rather complete understanding of some quasilinear elliptic problems on a bounded domain or over the entireR
 N
 .

Boundary blow-up solutions and their applications in quasilinear elliptic equations

Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian

The degenerate elliptic equation λ∆pu + uq−1(1 − ur) = 0 with zero Dirichlet boundary condition, where λ is a positive parameter, 2 < p < q and r > 0, is studied in three aspects: existence of maximal solution, λdependence of maximal solution and multiplicity of solutions.

https://www.ams.org/proc/2001-129-02/S0002-9939-00-05723-3/S0002-9939-00-05723-3.pdf

Positive solutions of a degenerate elliptic equation with logistic reaction

Multiplicity Result for a Degenerate Elliptic Equation with Logistic Reaction

In this paper we give two existence results for a class of degen- erate diusion equations with p-Laplacian. One is on a unique global strong solution, and the other is on a global attractor. It is also shown that the global attractor coincides with the unstable set of the set of all stationary solutions. As a by-product, an a-priori estimate for solutions of the corresponding elliptic equations is obtained.

/pdf/global-attractors-for-a-class-of-degenerate-diffusion-4g46ljumqu.pdf

Global attractors for a class of degenerate diffusion equations

Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized $\pi$ in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for $\pi$ by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transformation is also given.

Shingo Takeuchi

Papers

Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian

Positive solutions of a degenerate elliptic equation with logistic reaction

Multiplicity Result for a Degenerate Elliptic Equation with Logistic Reaction

Global attractors for a class of degenerate diffusion equations

A new form of the generalized complete elliptic integrals