Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula $$\left\| x\right\| _{\Phi ,s}=\inf _{k>0}\frac{1}{k}s\left( \int _T \Phi (kx)d\mu \right) $$ x Φ , s = inf k > 0 1 k s ∫ T Φ ( k x ) d μ , where s and $$\Phi $$ Φ are an outer and Orlicz function respectively and x is a measurable real-valued function over a $$\sigma $$ σ -finite measure space $$(T,\Sigma ,\mu )$$ ( T , Σ , μ ) . In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure $$\mu $$ μ is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

Multiple solutions for a quasilinear Schrödinger–Poisson system

In this article, we consider the following quasilinear Schrödinger–Poisson system $$ \textstyle\begin{cases} -\Delta u+V(x)u-u\Delta (u^{2})+K(x)\phi (x)u=g(x,u), \quad x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, \quad x\in \mathbb{R}^{3}, \end{cases} $$ { − Δ u + V ( x ) u − u Δ ( u 2 ) + K ( x ) ϕ ( x ) u = g ( x , u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ V , K : R 3 → R and $g:\mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ g : R 3 × R → R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.

Properties and Bounds of Jensen-Type Functionals via Harmonic Convex Functions

Dragomir introduced the Jensen-type inequality for harmonic convex functions (HCF) and Baloch et al. studied its different variants, such as Jensen-type inequality for harmonic h -convex functions. In this paper, we aim to establish the functional form of inequalities presented by Baloch et al. and prove the superadditivity and monotonicity properties of these functionals. Furthermore, we derive the bound for these functionals under certain conditions. Furthermore, we define more generalized functionals involving monotonic nondecreasing concave function as well as evince superadditivity and monotonicity properties of these generalized functionals.

Picking Sequences and Monotonicity in Weighted Fair Division

We study the problem of fairly allocating indivisible items to agents with different entitlements, which captures, for example, the distribution of ministries among political parties in a coalition government. Our focus is on picking sequences derived from common apportionment methods, including five traditional divisor methods and the quota method. We paint a complete picture of these methods in relation to known envy-freeness and proportionality relaxations for indivisible items as well as monotonicity properties with respect to the resource, population, and weights. In addition, we provide characterizations of picking sequences satisfying each of the fairness notions, and show that the well-studied maximum Nash welfare solution fails resource- and population-monotonicity even in the unweighted setting. Our results serve as an argument in favor of using picking sequences in weighted fair division problems.

Bounds for the generalized Marcum function of the second kind

In this paper, we focus on the generalized Marcum function of the second kind of order $$\nu >0$$ ν > 0 , defined by $$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{\nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$ R ν ( a , b ) = c a , ν a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 K ν - 1 ( a t ) d t , where $$a>0, b\ge 0,$$ a > 0 , b ≥ 0 , $$K_{\nu }$$ K ν stands for the modified Bessel function of the second kind, and $$c_{a,\nu }$$ c a , ν is a constant depending on a and $$\nu $$ ν such that $$R_{\nu }(a,0)=1.$$ R ν ( a , 0 ) = 1 . Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones.

Site-Monotonicity Properties for Reflection Positive Measures with Applications to Quantum Spin Systems

We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application, we derive site-monotonicity properties for the spin–spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates—improving previous positivity results which hold for the Cesàro sum. We also derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model and lattice permutations, thus extending the previous results of Lees and Taggi (2019).

Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels $( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )$ ( a − 1 A B R ∇ δ , γ y ) ( η ) of order $0<\delta <0.5$ 0 < δ < 0.5 , $\beta =1$ β = 1 , $0<\gamma \leq 1$ 0 < γ ≤ 1 starting at $a-1$ a − 1 . If $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y ) ( \eta )\geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 , then we deduce that $y(\eta )$ y ( η ) is $\delta ^{2}\gamma $ δ 2 γ -increasing. That is, $y(\eta +1)\geq \delta ^{2} \gamma y(\eta )$ y ( η + 1 ) ≥ δ 2 γ y ( η ) for each $\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}$ η ∈ N a : = { a , a + 1 , … } . Conversely, if $y(\eta )$ y ( η ) is increasing with $y(a)\geq 0$ y ( a ) ≥ 0 , then we deduce that $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta ) \geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 . Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.