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

Abstract

and Applied Analysis 3 Lemma 2.1. If α ∈ 0, 1 , then 1 2α log 2 − logα > 3 log 2. Proof. For α ∈ 0, 1 , let f α 1 2α log 2 − logα , then simple computations lead to f ′ α 2 ( log 2 − 1 − 2 logα − 1 α , 2.1 f ′′ α 1 α2 1 − 2α . 2.2 From 2.2 we clearly see that f ′′ α > 0 for α ∈ 0, 1/2 , and f ′′ α < 0 for α ∈ 1/2, 1 . Then from 2.1 we get f ′ α ≤ f ′ ( 1 2… (More)

Topics

Figures and Tables

Sorry, we couldn't extract any figures or tables for this paper.

Slides referencing similar topics