For \(p\in (1,+\infty )\) , let \(q=p/(p-1)\) . This paper deals mostly with q-arclength of a p-ellipse, where the p-ellipse is the analogue of the ellipse, and q-arclength is the arclength under the \(l_{q}\) -metric. The authors present the representation of q-arclength of the p-ellipse in terms of Gaussian hypergeometric function by use of the generalized trigonometric functions. With the development of its related generalized elliptic integrals and bivariate means, several sharp upper and lower bounds of q-arclength of the p-ellipse will be proved, which extend some previous well-known results.
