The monotonicity results and sharp inequalities for some power-type means of two arguments

2012/10/24 by Zhen-Hang Yang, Yang, Zhen-Hang
Mathematics · #26D05 (Primary ) 26A48 (Secondary) #26E60 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications #math.CA #msc:26A48 #msc:26D05 #msc:26E60

paper · pdf · doi:10.48550/arxiv.1210.6478

11 pages

arxiv created 2012/10/24 · openalex publication_date 2012/10/24 · arxiv updated 2012/10/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For a,b>0 with a≠ b, we define M_p=M^1/p(a^p,b^p)ifp≠ 0 and M₀=√(ab), where M=A,He,L,I,P,T,N,Z and Y stand for the arithmetic mean, Heronian mean, logarithmic mean, identric (exponential) mean, the first Seiffert mean, the second Seiffert mean, Neuman-Sándor mean, power-exponential mean and exponential-geometric mean, respectively. Generally, if M is a mean of a and b, then M_p is also, and call "power-type mean". We prove the power-type means P_p, T_p, N_p, Z_p are increasing in p on ℝ and establish sharp inequalities among power-type means A_p, He_p, L_p, I_p, P_p, N_p, Z_p, Y_p% . From this a very nice chain of inequalities for these means L₂<P<N_1/2<He<A_2/3<I<Z_1/3<Y_1/2 follows. Lastly, a conjecture is proposed.

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