Gagliardo–Nirenberg inequalities in logarithmic spaces

Logarithmic Harnack inequalities∗

Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9, 12]. Recently, Diaconis and Saloff-Coste [8] considered logarithmic Sobolev inequalities for Markov chains. The lower bounds for log-Sobolev constants can be used to improve convergence bounds for ...

Modified logarithmic Sobolev inequalities and transportation inequalities

Inequalities for Generalized Logarithmic Means

For p ∈ R, the generalized logarithmic mean Lp of two positive numbers a and b is defined as Lp a, b a, for a b, LP a, b b 1 − a 1 / p 1 b − a 1/p , for a/ b, p / − 1, p / 0, LP a, b b − a / log b − loga , for a/ b, p −1, and LP a, b 1/e b/a 1/ b−a , for a/ b, p 0. In this paper, we prove that G a, b H a, b 2L−7/2 a, b , A a, b H a, b 2L−2 a, b , and L−5 a, b H a, b for all a, b > 0, and the co...

Inequalities Among Logarithmic-Mean Measures

In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, logarithmic means, etc. Inequalities involving logarithmic mean with differences among other means are presented.

Optimal inequalities for the power, harmonic and logarithmic means

For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...