A Rational Approximation for the Complete Elliptic Integral of the First Kind

Abstract

Let K ( r ) be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for K ( r ) . More precisely, we establish the inequality 2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 ( r ′ ) 2 + 110 r ′ + 21 for r ∈ ( 0 , 1 ) , where r ′ = 1 − r 2 . The lower bound is sharp.

A new chain of inequalities involving the Toader-Qi, logarithmic and exponential means

“…Such type power series appear frequently in certain special functions. A similar signs rule was proven in [36,39] and plays a key role in the study of means and special functions, see for example, [21,28,31,34,37,43].…”