Inequalities and bounds for elliptic integrals

Abstract

Computable lower and upper bounds for the symmetric elliptic integrals and for Legendre's incomplete integral of the first kind are obtained. New bounds are sharper than those known earlier. Several inequalities involving integrals under discussion are derived.

A power mean inequality involving the complete elliptic integrals

“…The complete elliptic integrals play a very important role in the study of conformal invariants [7], quasiconformal mappings [5,6,7,12] and Ramanujan's modular equations [4]. Numerous sharp inequalities and elementary approximations for the complete elliptic integrals have been proved in [2,3,7,8,13,14]. The special function m(r) is defined as…”

Optimal inequalities for a Toader-type mean by quadratic and contraharmonic means

“…For p ∈ R and a, b > 0 with a = b, the pth power mean M p (a, b) [7,9,17,18,22,32,35,37,38], pth Lehmer mean L p (a, b) [27,34], harmonic mean H(a, b), geometric mean G(a, b), arithmetic mean A(a, b), Toader mean T (a, b) [10,14,16,28], centroidal mean C(a, b) [6,36], quadratic mean Q(a, b) [19], contraharmonic mean C(a, b) [5,13] are, respectively, defined by The Toader mean T (a, b) has been well known in the mathematical literature for many years (see [20,21,24]), which is related to the complete elliptic integral of the second kind E(r) = π/2 0 (1 − r 2 sin 2 θ) 1/2 dθ(r ∈ (0, 1)) [12,15,25,30,31,33,39,40] and it can be rewritten as…”

Optimal bounds for a Toader-type mean in terms of one-parameter quadratic and contraharmonic means

“…stands for the symmetric complete elliptic integral of the second kind [7,8,10], therefore it can't be expressed in terms of the elementary transcendental functions.…”