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The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

Authors
Publisher
Abstract and Applied Analysis
Publication Date
Disciplines
  • Mathematics

Abstract

For 𝑝 ∈ ℝ , the power mean 𝑀 𝑝 ( 𝑎 , 𝑏 ) of order 𝑝 , logarithmic mean 𝐿 ( 𝑎 , 𝑏 ) , and arithmetic mean 𝐴 ( 𝑎 , 𝑏 ) of two positive real values 𝑎 and 𝑏 are defined by 𝑀 𝑝 ( 𝑎 , 𝑏 ) = ( ( 𝑎 𝑝 + 𝑏 𝑝 ) / 2 ) 1 / 𝑝 , for 𝑝 ≠ 0 and 𝑀 𝑝 √ ( 𝑎 , 𝑏 ) = 𝑎 𝑏 , for 𝑝 = 0 , 𝐿 ( 𝑎 , 𝑏 ) = ( 𝑏 − 𝑎 ) / ( l o g 𝑏 − l o g 𝑎 ) , for 𝑎 ≠ 𝑏 and 𝐿 ( 𝑎 , 𝑏 ) = 𝑎 , for 𝑎 = 𝑏 and 𝐴 ( 𝑎 , 𝑏 ) = ( 𝑎 + 𝑏 ) / 2 , respectively. In this paper, we answer the question: for 𝛼 ∈ ( 0 , 1 ) , what are the greatest value 𝑝 and the least value 𝑞 , such that the double inequality 𝑀 𝑝 ( 𝑎 , 𝑏 ) ≤ 𝛼 𝐴 ( 𝑎 , 𝑏 ) + ( 1 − 𝛼 ) 𝐿 ( 𝑎 , 𝑏 ) ≤ 𝑀 𝑞 ( 𝑎 , 𝑏 ) holds for all 𝑎 , 𝑏 > 0 ?

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