Application of Quasi Subordination Associated with Generalized Sakaguchi Type Functions

In this article, a new class of analytic functions which is defined by terms of a quasi-subordination is introduced. The coefficient estimates, including the classical inequality of functions belonging to this class, are then derived. Also, several special improving results for the associated classes involving the subordination are presented.

On the Existence of Extremals for Moser-Type Inequalities in Gauss Space

The existence of an extremal in an exponential Sobolev-type inequality, with optimal constant, in Gauss space is established. A key step in the proof is an augmented version of the relevant inequality, which, by contrast, fails for a parallel classical inequality by Moser in the Euclidean space.

Noncommutative strong maximals and almost uniform convergence in several directions

Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$ -norm of the $\limsup $ of a sequence of operators as a localized version of a $\ell _\infty /c_0$ -valued $L_p$ -space. In particular, our main result gives a strong $L_1$ -estimate for the $\limsup $ —as opposed to the usual weak $L_{1,\infty }$ -estimate for the $\mathop {\mathrm {sup}}\limits $ —with interesting consequences for the free group algebra. Let $\mathcal{L} \mathbf{F} _2$ denote the free group algebra with $2$ generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F} _2)$ for which the free Poisson semigroup converges to the initial data. Currently, the best known result is $L \log ^2 L(\mathcal{L} \mathbf{F} _2)$ . We improve this result by adding to it the operators in $L_1(\mathcal{L} \mathbf{F} _2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative $\limsup $ together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak $(\Phi ,\Phi )$ inequality—as opposed to weak $(\Phi ,1)$ —for noncommutative multiparametric martingales and $\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$ . This logarithmic power is an $\varepsilon $ -perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

Lower bounds for Mahler measure that depend on the number of monomials

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients and the number of monomials. In one variable, our result generalizes a classical inequality of Mahler. In [Formula: see text] variables, our result depends on [Formula: see text] as an ordered group, and in general, our lower bound depends on the choice of ordering.

Dynamics and eigenvalues in dimension zero

Let $X$ be a compact, metric and totally disconnected space and let $f:X\rightarrow X$ be a continuous map. We relate the eigenvalues of $f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$ to dynamical properties of $f$, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0, 1 is an eigenvalue. This stands in contrast with a classical inequality of Manning that bounds the entropy of $f$ below by the spectral radius of $f_{\ast }$.

Penalty-Based and Other Representations of Economic Inequality

Economic inequality measures are employed as a key component in various socio-demographic indices to capture the disparity between the wealthy and poor. Since their inception, they have also been used as a basis for modelling spread and disparity in other contexts. While recent research has identified that a number of classical inequality and welfare functions can be considered in the framework of OWA operators, here we propose a framework of penalty-based aggregation functions and their associated penalties as measures of inequality.

Intersection numbers in the curve complex via subsurface projections

A classical inequality which is due to Lickorish and Hempel says that the distance between two curves in the curve complex can be measured by their intersection number. In this paper, we show a converse version; the intersection number of two curves can be measured by the sum of all subsurface projection distances between them. As an application of this result, we obtain a coarse decreasing property of the intersection numbers of the multicurves contained in tight multigeodesics. Furthermore, by using this property, we give an algorithm for determining the distance between two curves in the curve complex. Indeed, such algorithms have also been found by Birman–Margalit–Menasco, Leasure, Shackleton, and Webb: we will briefly compare our algorithm with some of their algorithms, for detailed quantitative comparison of all known algorithms including our algorithm, we refer the reader to the paper of Birman–Margalit–Menasco [1].

Inequalities for eigenvalues of compact operators in a Hilbert space

Let [Formula: see text] be a compact operator in a separable Hilbert space and [Formula: see text] be the eigenvalues of [Formula: see text] with their multiplicities enumerated in the non-increasing order of their absolute values. We prove the inequality [Formula: see text] where [Formula: see text] and [Formula: see text] are the singular values of [Formula: see text] and of [Formula: see text], respectively, enumerated with their multiplicities in the non-increasing order. This result refines the classical inequality [Formula: see text]

A SHARP VERSION OF BONSALL’S INEQUALITY

The best possible constant in a classical inequality due to Bonsall is established by relating that inequality to Young’s. Further, this extends the range of Bonsall’s inequality and yields a reverse inequality. It also provides a better constant in an inequality of Hardy, Littlewood and Pólya.