  The elliptic integral and its various generalizations are playing very
important and basic role in different branches of modern mathematics. It is
well known that they cannot be represented by the elementary transcendental
functions. Therefore, there is a need for sharp computable bounds for the
family of integrals. In this paper, by virtue of two new tools, we study
monotonicity and convexity of certain combinations of the complete elliptic
integrals of the first kind, and obtain new sharp bounds and inequalities for
them. In particular, we prove that the function $\mathcal{K}\left( \sqrt{%
x}\right) /\ln \left( c/\sqrt{1-x}\right) $ is concave on $\left( 0,1\right) $
if and only if $c=e^{4/3}$, where $\mathcal{K}$ denotes the complete elliptic
integrals of the first kind.
