• Journal of the Chungcheong Mathematical Society
• /
• v.27 no.2
• /
• pp.165-172
• /
• 2014

In this paper, we derive some valuable inequalities of Rado's and Poponov's types on the open interval of positive real numbers, and then show weighted generalizations of Rado's and Poponov's inequalities on the set of positive real numbers equipped with compactly supported probability measure.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.6
• /
• pp.1981-1990
• /
• 2017

Let R be a root system in $\mathbb{R}^d$ with Coxeter-Weyl group W and let k be a nonnegative multiplicity function on R. The generalized volume mean of a function $f{\in}L^1_{loc}(\mathbb{R}^d,m_k)$, with $m_k$ the measure given by $dmk(x):={\omega}_k(x)dx:=\prod_{{\alpha}{\in}R}{\mid}{\langle}{\alpha},x{\rangle}{\mid}^{k({\alpha})}dx$, is defined by: ${\forall}x{\in}\mathbb{R}^d$, ${\forall}r$ > 0, $M^r_B(f)(x):=\frac{1}{m_k[B(0,r)]}\int_{\mathbb{R}^d}f(y)h_k(r,x,y){\omega}_k(y)dy$, where $h_k(r,x,{\cdot})$ is a compactly supported nonnegative explicit measurable function depending on R and k. In this paper, we prove that for almost every $x{\in}\mathbb{R}^d$, $lim_{r{\rightarrow}0}M^r_B(f)(x)= f(x)$.