The optimal power mean bounds for two convex combinations of A-G-H means
نویسندگان
چکیده
منابع مشابه
Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means
We find the greatest values $alpha_{1} $ and $alpha_{2} $, and the least values $beta_{1} $ and $beta_{2} $ such that the inequalities $alpha_{1} C(a,b)+(1-alpha_{1} )H(a,b)
متن کاملoptimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means
we find the greatest values $alpha_{1} $ and $alpha_{2} $, and the least values $beta_{1} $ and $beta_{2} $ such that the inequalities $alpha_{1} c(a,b)+(1-alpha_{1} )h(a,b)
متن کاملOptimal Convex Combinations Bounds of Centroidal and Harmonic Means for Weighted Geometric Mean of Logarithmic and Identric Means
In this paper, optimal convex combination bounds of centroidal and harmonic means for weighted geometric mean of logarithmic and identric means are proved. We find the greatest value λ(α) and the least value Δ(α) for each α ∈ (0,1) such that the double inequality: λC(a,b)+(1−λ)H(a,b) < Lα (a,b)I1−α (a,b) < ΔC(a,b)+(1−Δ)H(a,b) holds for all a,b > 0 with a = b. Here, C(a,b), H(a,b) , L(a,b) and I...
متن کاملOptimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means
and Applied Analysis 3 Theorem 1.1. The double inequality α1H a, b 1 − α1 Q a, b < M a, b < β1H a, b ( 1 − β1 ) Q a, b 1.7 holds for all a, b > 0with a/ b if and only if α1 ≥ 2/9 0.2222 . . . and β1 ≤ 1−1/ √ 2 log 1 √ 2 0.1977 . . . . Theorem 1.2. The double inequality α2G a, b 1 − α2 Q a, b < M a, b < β2G a, b ( 1 − β2 ) Q a, b 1.8 holds for all a, b > 0with a/ b if and only if α2 ≥ 1/3 0.3333...
متن کاملThe Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means
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 ) 4 ( log 2 − 1 < 0 2.3 for α ∈ 0, 1 ....
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Mathematical Inequalities
سال: 2012
ISSN: 1846-579X
DOI: 10.7153/jmi-06-03