We establish the embedding of the critical Sobolev-Lorentz-Zygmund space $H_{p,q,\lambda ₁,...,\lambda ₘ}^{n/p}\left(ℝⁿ\right)$ into the generalized Morrey space ${ℳ}_{\Phi ,r}\left(ℝⁿ\right)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H_{p,q,\lambda ₁,...,\lambda ₘ}^{n/p+1}\left(ℝⁿ\right)$. O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.

There are many relations involving the geometric means $G_n\left(x\right)$ and power means ${\left[A_n\left(x^{\gamma }\right)\right]}^{1/\gamma }$ for positive $n$-vectors $x$. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1-\lambda )G_n^{\gamma }\left(x\right)+\lambda A_n^{\gamma }\left(x\right)\ge A_n\left(x^{\gamma }\right)$ and $(1-\lambda )G_n^{\gamma }\left(x\right)+\lambda A_n^{\gamma }\left(x\right)\le A_n\left(x^{\gamma }\right)$ with parameters $\lambda \in ℝ$ and $\gamma \in (0,1).$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated....

We provide a set of optimal estimates of the form (1-μ)/𝓐(x,y) + μ/ℳ (x,y) ≤ 1/ℬ(x,y) ≤ (1-ν)/𝓐(x,y) + ν/ℳ (x,y) where 𝓐 < ℬ are two of the Seiffert means L,P,M,T, while ℳ is another mean greater than the two.

2000 Mathematics Subject Classification: 26D10, 26D15.Here are presented Ostrowski type inequalities over spherical shells. These regard sharp or close to sharp estimates to the difference of the average of a multivariate function from its value at a point.

Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral ${\int }_a^{b}f\left(e^{it}\right)du\left(t\right)$ of continuous complex valued integrands $f:𝒞\left(0,1\right)\to ℂ$ defined on the complex unit circle $𝒞\left(0,1\right)$ and various subclasses of integrators $u:\left[a,b\right]\subseteq \left[0,2\pi \right]\to ℂ$ of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well.