Convex symmetrization and Pólya–Szegö inequality

We prove a Pólya–Szegö inequality involving a convex symmetrization of functions and we investigate the equality case.     

A generalization of the Pólya–Szegö inequality for one-dimensional functionals

We consider the Pólya–Szegö type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner’s symmetrization.     

On the Mixed Pólya–Szeg? Principle

In this paper,the mixed Pólya-Szeg? principle is established.By the mixed Pólya-Szeg? principle,the mixed Morrey-Sobolev inequality and some new analytic inequalities are obtained.     

A generalization of the Pólya–Vinogradov inequality

In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya–Vinogradov inequality. We show how the Dobrowolski–Williams approach is related to the classical proof of Pólya–Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65–71, 2001). In passing, we also obtain sharper explicit versions of the Pólya–Vinogradov inequality.     

An asymmetric affine Pólya–Szeg? principle

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya–Szeg? symmetrization principle for functions on \mathbb Rⁿ{\mathbb R^n} . Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which is stronger than its classical Euclidean counterpart.     

Generalized Pólya–Szegö type inequalities for some non-commutative geometric means

In this paper, we shall show generalized Pólya–Szegö type inequalities of n positive invertible operators on a Hilbert space for any integer $n?3$ in terms of the following two typical non-commutative geometric means, that is, one is the higher order weighted geometric mean due to Lawson–Lim which is an extension of the Ando–Li–Mathias geometric mean, and the other is the weighted chaotic geometric mean. Among others, the Specht ratio plays an important role in our discussion, which is the upper bound of a ratio type reverse of the weighted arithmetic–geometric mean inequality.     

Stability of Pólya–Szegő inequality for log-concave functions

A quantitative version of Pólya–Szeg? inequality is proven for log-concave functions in the case of Steiner and Schwarz rearrangements.     

On One Generalization of the Hardy–Littlewood–Pólya Inequality

We generalize the Hardy–Littlewood–Pólya inequality for numerical sets to certain sets of vectors on a plane.     

On the shape of random Pólya structures

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton–Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n\left(v\right)$, that is attached to a node $v$ in $C_n$, is maximally of size $\left|F_n\left(v\right)\right|=O(logn)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities.In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\left|F_n\left(v\right)\right|$, namely $\left|F_n\left(v\right)\right|=\Theta (logn)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n\left(v\right)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.     

On a theorem of G. Pólya

f(z), :f(n)=0 (n=0, ±1, ±2, ...). ((n)} L _p ,p>1, .     

On a reverse Heinz–Kato–Furuta inequality

In the set up of Minkowski spaces, the Schwarz inequality holds with the reverse inequality sign. As a consequence, the same occurs with the triangle inequality. In this note, extensions of this indefinite version of the Schwarz inequality are presented. Namely, a reverse Heinz–Kato–Furuta inequality valid for timelike vectors is included and related inequalities that also hold with the reverse sign are investigated.     

Equivalence between Pólya–Szegő and relative capacity inequalities under rearrangement

The transformations of functions acting on sublevel sets that satisfy a Pólya–Szeg? inequality are characterized as those being induced by transformations of sets that do not increase the associated capacity.     

On the Pólya–Szégö Inequality for Functionals with Variable Exponent

Analogues of the Pólya–Szégö inequality with variable exponent in the integrand are considered. Necessary and sufficient conditions for the fulfillment of these inequalities are obtained.     

On the reverse Orlicz Busemann–Petty centroid inequality

In this paper, the Orlicz centroid body, defined by E. Lutwak, D. Yang and G. Zhang, and the extrema of some affine invariant functionals involving the volume of the Orlicz centroid body are investigated. The reverse form of the Orlicz Busemann–Petty centroid inequalities is obtained in the two-dimensional case.