On approximating the complete p -elliptic integral of the second kind by weighted Lehmer means

A known sharp double inequality for complete elliptic integral of the second kind <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">E</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> E(r) by Lehmer mean is extended to the complete p -elliptic integral of the second kind <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">E</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> Eₚ(r) . For p ∈ [2, 5], sharp approximations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">E</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> Eₚ(r) are established by weighted Lehmer means. This result confirms the validity of [T. Zhao and M. Wang, Discrete approximation of complete p-elliptic integral of the second kind and its application , Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 118 (2024), no. 1, 18, Conjecture 4.6] when p = 4 and provides new refinements of the Lehmer mean inequality for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">E</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> E(r) . Additionally, for p ∈ [ p 0 , p 1 ] (where p 0 ≈ 1.30995 and p 1 ≈ 11.0205), more precise bounds are derived by exploiting the absolute monotonicity of a constructed function.