Sentence examples for mean inequality and from inspiring English sources

by the arithmetic mean-geometric mean inequality and the right of inequality (4.1).

From inequality (2.4) and in view of the following mean inequality and inverse Hölder's inequality [10, page 24], we have.

end{aligned} From the equality conditions of the arithmetic-geometric mean inequality and (h K,u =rho K,u)), the equality holds if and only if K is the unit ball centered at the origin.

Second, some inequalities were proved for mappings (q th powers of first derivatives belonging to class (Q_{s}(I)) using the Čebyšev's inequality, H ölder inequality, Power mean inequality, and some other classical inequalities.

Using the geometric arithmetic mean inequality and the divergence theorem, we have begin{aligned} P(B &= n omega _n = nint _{B} dy = n int _E (mathrm{det}, nabla T ^{1/n} dx = n int _E (lambda _1 cdots lambda _n)^{1/n}dx &le int _E (lambda _1 + cdots + lambda _n dx = int _E text {div},T,dx = int _{partial E} T cdot nu ^E d{mathcal H}^{n-1} le P E).

In addition, they also refined the μ-weighted arithmetic-harmonic mean inequality and extended it to an operator version as follows: begin{aligned} &anabla_{mu}bge a!_{mu}b+ 2r (anabla b-a b ), end{aligned} (1.3) b-a b{alignend{aligned_{mu}Bge A!_{mu}B+2r (Anabla B-A!B), end{aligned} (1.3) where (a,begin{alignedathcal{B}^(H) ), (muin[0,1]), and (r=min { {mu,1-mu} }).

In this paper, we employ iteration on operator version of the famous Young inequality and obtain more arithmetic-geometric mean inequalities and the reverse versions for positive operators.

This paper aims to provide a method to obtain more arithmetic-geometric mean inequalities and the reverse version for positive operators.

From the hypotheses and by Jensen's inequality, the means inequality, and inverse Hölder's inequality, we obtain that.

In view of the means inequality and integrating two sides of (2.10) over from to and noticing Hölder integral inequality, we observe that (2.11).

Hence, from (2.2) and (2.3) and in view of the arithmetic-geometric means inequality and Hölder inequality with indices μ and λ, it follows that u ( x, y ) l ≤ 1 2 u ( x, y ) l - 1 ∫ a i b i ∫ c i d i ∂ 2 ∂ s i ∂ t i u ( x, y ; s i, t i ) d s i d t i ≤ 1 2 u ( x, y ) l - 1 [ ( b i - a i ) ( c i - d i ) ] 1 / μ ∫ a i x i ∫ c i y i ∂ 2 ∂ s i ∂ t i u ( x, y ; s i, t i ) λ d s i d t i 1 / λ. (2.4).