Related papers: The monotonicity results and sharp inequalities fo…

Let $\mu_p(A,B,t)=(tA^p+(1-t)B^p)^{1/p}$ denote the weighted power mean between positive operators $A$ and $B$. We show that the function $t\to \|A-\mu_p(A,B,t)\|_2$ is monotonically decreasing whenever $1/2 \leq p \leq 1$. Hence showing…

We prove that whenever the selfmapping $(M_1,\dots,M_p)\colon I^p \to I^p$, ($p \in \mathbb{N}$ and $M_i$-s are $p$-variable means on the interval $I$) is invariant with respect to some continuous and strictly monotone mean $K \colon I^p…

The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where…

In this paper, we prove that the double inequality $M_{p}(a,b)<B(a,b)<M_{q}(a,b)% holds for all $a, b>0$ with $a\neq b$ if and only if $p\leq 4\log 2/(4+2\log 2-\pi)=1.2351\cdots$ and $q\geq 4/3$, where $%…

In the paper, by establishing the monotonicity of some functions involving the sine and cosine functions, the authors provide concise proofs of some known inequalities and find some new sharp inequalities involving the Seiffert,…

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $\Sigma$. Fix a number $1 < p < 3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become…

Let m(a,b) and M(a,b,c) be symmetric means. We say that M is type 1 invariant with respect to m if M(m(a,c),m(a,b),m(b,c)) = M(a,b,c) for all a, b, c > 0. If m is strict and isotone, then we show that there exists a unique M which is type 1…

In this paper we prove and discuss some new $\left( H_{p},L_{p}\right)$ type inequalities of maximal operators of $T$ means with respect to the Vilenkin systems with monotone coefficients. We also apply these inequalities to prove strong…

Each family $\mathcal{M}$ of means has a natural, partial order (point-wise order), that is $M \le N$ iff $M(x) \le N(x)$ for all admissible $x$. In this setting we can introduce the notion of interval-type set (a subset $\mathcal{I}…

We discuss $(H_p,L_p)$ and $(H_p,\text{weak}-L_p)$ type inequalities of weighted maximal operators of $T$ means with respect to the Vilenkin systems with monotone coefficients, considered in \cite{tut4} and prove that these results are the…

e study the properties of the mean-type mappings ${\bf M}\colon I^p \to I^p$ of the form $${\bf M}(x_1,\dots,x_p):=\big(M_1(x_{\alpha_{1,1}},\dots,x_{\alpha_{1,d_1}}),\dots,M_p(x_{\alpha_{p,1}},\dots,x_{\alpha_{p,d_p}})\big),$$ where $p$…

On the setting of positive definite operators we study the near-order properties of power means such as the quasi-arithmetic mean (H\"{o}lder mean) and R\'{e}nyi power mean. We see the monotonicity of spectral geometric mean and Wasserstein…

In this paper authors establish the two sided inequalities for the following two new means $$X=X(a,b)=Ae^{G/P-1},\quad Y=Y(a,b)=Ge^{L/A-1}.$$ As well as many other well known inequalities involving the identric mean $I$ and the logarithmic…

Motivated by the work of Anderson, Vamanamurthy and Vuorinen \cite{avv}, in this paper authors study the log-convexity and log-concavity of Power mean, Identric mean, weighted Power mean, Lehmer mean, Modified Alzer mean, and establish the…

Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\geq0$ and $0<p\leq1$, then \[\Vert N\Vert_{p,\infty}\leq2\Vert…

We prove some sufficient conditions implying $l^p$ inequalities of the form $||x||_p \leq ||y||_p$ for vectors $ x, y \in [0,\infty)^n$ and for $p$ in certain positive real intervals. Our sufficient conditions are strictly weaker than the…

Recently, Nagy-To\'okos and Totik-Varga proved an asymptotically sharp $L^{p}$ Bernstein type inequality on union of finitely many intervals. We extend this inequality to the case when the power $p$ is between $0$ and $1$; such sharp…

As non-commutative versions of the quasi-arithmetic mean, we consider the Lim-P\'{a}lfia's power mean, R\'{e}nyi right mean and R\'{e}nyi power means. We prove that the Lim-P\'{a}lfia's power mean of order $t \in [-1,0)$ is weakly…

For a,b>0 with a\not=b, let N(a,b) denote the Neuman-S\'andor mean defined by N(a,b)=((a-b)/(2arcsinh((a-b)/(a+b)))) and A_{r}(a,b) denote the r-order power mean. We present the sharp power means bounds for the Neuman-S\'andor mean:…

For a,b>0 with a\not=b, let T(a,b) denote the second Seiffert mean defined by T(a,b)=((a-b)/(2arctan((a-b)/(a+b)))) and A_{r}(a,b) denote the r-order power mean. We present the sharp bounds for the second Seiffert mean in terms of power…