Absolute Monotonicity Involving the Complete Elliptic Integrals of the First Kind with Applications

Abstract

Let K (r) be the complete elliptic integrals of the first kind for r ∈ (0, 1) and ${f_p}(x) = [{(1 - x)^p}{\cal K}(\sqrt x )]$. Using the recurrence method, we find the necessary and sufficient conditions for the functions ∔ f′_p, ln f_p, — (ln f_p(i = 1, 2, 3) to be absolutely monotonic on (0, 1). As applications, we establish some new bounds for the ratios and the product of two complete integrals of the first kind, including the double inequalities

$${f_p}(x) = [{(1 - x)^p}{\cal K}(\sqrt x )]$$

for r ∈ (0, 1) and p ≥ 13/32, where $\matrix{{{{{\rm{exp}}\left[ {{r^2}\left( {1 - {r^2}} \right)/64} \right]} \over {{{\left( {1 + r} \right)}^{1/4}}}} < {{{\cal K}\left( r \right)} \over {{\cal K}\left( {\sqrt r } \right)}} < \exp \left[ { - {{r(1 - r)} \over 4}} \right],} \hfill \cr {{\pi \over 2}\exp \left[ {{\theta _0}\left( {1 - 2{r^2}} \right)} \right] < {\pi \over 2}{{{\cal K}\left( {{r^\prime }} \right)} \over {{\cal K}\left( r \right)}} < {\pi \over 2}{{\left( {{{{r^\prime }} \over r}} \right)}^p}\exp \left[ {{\theta _p}\left( {1 - 2{r^2}} \right)} \right],} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\cal K}^2}\left( {{1 \over {\sqrt 2 }}} \right) \le {\cal K}\left( r \right){\cal K}\left( {{r^\prime }} \right) \le {1 \over {\sqrt {2r{r^\prime }} }}{{\cal K}^2}\left( {{1 \over {\sqrt 2 }}} \right)} \hfill \cr }$ and θ_P = 2Γ (3/4)⁴ /ρ² − p.