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We present some monotonicity, convexity, and inequalities for the Agard distortion function ηK(t) and improve some well-known results.

For p∈[0,1], the generalized Seiffert mean of two positive numbers a and b is defined by Sp(a,b)=p(a-b)/arctan[2p(a-b)/(a+b)], 00 with a≠b, and give new bounds for the complete elliptic integrals of the second kind. Here, T(a,b)=(2/π)∫0π/2a2cos⁡2θ+b2sin⁡2θdθ denotes the Toader mean of two positive numbers a and b.