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

Abstract A known sharp double inequality for complete elliptic integral of the second kind E (r) E (r) by Lehmer mean is extended to the complete p -elliptic integral of the second kind E p (r) E₏ (r). For p ∈ 2, 5, sharp approximations of E p (r) E₏ (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 E (r) 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.