SIAM J. MATH. ANAL. Vol. 21, No. 2, pp. 536-549, March 1990 (C) 1990 Society for Industrial and Applied Mathematics 014 FUNCTIONAL INEQUALITIES FOR COMPLETE ELLIPTIC INTEGRALS AND THEIR RATIOS* G. D. ANDERSON,f M. K. VAMANAMURTHY, AND M. VUORINEN Abstract. Some functional inequalities satisfied by complete elliptic integrals of the first kind are obtained. These inequalities are sharp and generalize the functional identity of Landen. A related inequality is given for certain quotients of such integrals. Key words, complete elliptic integral, quasiconformal mapping, functional inequality, Teichmiiller ring, GriStzsch ring AMOS(MOS) subject classifications, primary 33A25, 33A70; secondary 30C60 1. Introduction. For 0 < r < 1 the functions (1.2) E’(r) E(r’), r’=.,/1- r 2 /2 (1.1) K(r)= (1-r-sinEt)-’/dt, K’(r)=K(r’), r’=/i-r2, E(r) (1 1.2 sin 2 t) 1/2 dt, d0 are known as complete elliptic integrals of the first and second kind, respectively [BF], [Bo], [BB2], and their values are listed in standard tables (e.g., [AS], [Fr]). The special combinations 7r K r 2 zr 1 (1.3) /z(r)- 2 K(r)’ y(s)- r-- /x(r) s are particularly important in quasiconformal analysis (cf. [LV]). The elliptic integral K(r) satisfies the following basic identities due to Landen [BF, = 163.01,164.02] (cf. [WW, p. 507]): ( 2x/ (i-r) 1 ,( (1.4) K\l+r]=(l+r)K(r), K =(l+r)K r), while the function/z(r) satisfies the identities 2 /x(r,/z (l-r) r 2 kc(r)---2/z (2x/ /x(r)/z(/1-r2)= ’ r =--’ l+r/ (1.5) (cf. [LV, Forms. (2.7), (2.9), (2.3), pp. 60, 61]). The first identity in (1.5) follows directly from definition (1.1), while the other two follow from (1.4). It is also well known that (1.6) log 1 4 </x (r) < log /" r for 0< r < 1 [LV, Form. (2.10), p. 61]. * Received by the editors January 25, 1988; accepted for publication (in revised form) March 7, 1989. t Michigan State University, East Lansing, Michigan 48824. This author’s work was supported in part by a grant from the National Science Foundation and in part by a grant from the Academy of Finland. t University of Auckland, Auckland, New Zealand. This author’s work was supported in part by a grant from the Academy of Finland. University of Helsinki, Helsinki, Finland. This author’s work was supported in part by the Alexander yon Humboldt Foundation. 536