Some inequalities for characteristic functions

In a recent paper, Matysiak and Szablowski [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] posed an interesting conjecture about a lower bound of real-valued characteristic functions. Under a suitable moment condition on distributions, we prove the conjecture to be true. The unified approach proposed here enables us to obtain new inequalities for characteristic functions. We also show by example that the improvement in the bounds is significant if more information about the distribution is available.     

Some inequalities for multivariate characteristic functions

We obtain some new lower and upper bounds for characteristic functions of multivariate distributions that can be useful in various applications. Supported by the Russian Foundation for Basic Research (grant Nos. 97-01-00273, 98-01-00621, and 98-01-00926) and by INTAS-RFBR (grant No. IR-97-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Some inequalities concerning the James constant in Banach spaces

Let A₂(X) be the constant introduced by Baronti, Casini and Papini. This paper discusses the constant A₂(X) and states an estimate in terms of the James constant. The estimate enables us to improve an inequality between the James and von Neumann-Jordan constants.     

Some norm inequalities concerning generalized inverses, 2

We extend to the von Neumann–Schatten classes C_p and norms ·_p, where 2  p < ∞, Penrose’s result on minimizing AXB − C₂. We give an example to show that this extension does not hold for 1  p < 2. The proof of the global inequality depends on local considerations.     

Some inequalities for characteristic functions. II

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] about a lower bound of real-valued characteristic functions. In this paper, applying an alternative approach we are able to give explicitly the ranges of argument for which the obtained inequalities hold true for general characteristic functions.     

Some relations between certain inequalities concerning analytic and univalent functions

By making use of the well-known assertions in [S.S. Miller, P.T. Mocanu, Differential Subordinations, Theory and Applications, Marcel Dekker, New York-Basel, 2000] and [M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (7) (1993) 234–237], several useful results of certain inequalities dealing with analytic and univalent functions are obtained and then certain consequences of them are also indicated.     

Some results concerning displacements in a non-homogeneous elastic layer

Résumé On sait qu'un semi-espace de terrain isotrope, élastique et incompressible — dont le module de cisaillement accroît linéairement avec la profondeur, partant de zéro à la surface — réagit à la charge superficielle exactement selon les hypothèses faites par Winkler dans sa théorie de la réaction du niveau de fondation. On examine ici jusqu'à quel point la validité de ce résultat s'applique à un milieu de profondeur limitée. On y démontre que, si la profondeur de la couche dépasse la largeur de la surface chargée, le tassement d'une charge répartie uniformément est aussi pratiquement uniforme. Cependant, au delà du sol chargé, la surface se souleve d'une manière sensible, même lorsque la couche est d'une épaisseur modérée.     

Some inequalities derived from log-Sobolev inequalities

We show how a log-Sobolev inequality can be manipulated to give an improved version of itself.

     

Some deviation inequalities

We introduce a concentration property for probability measures onR ⁿ , which we call Property (); we show that this property has an interesting stability under products and contractions (Lemmas 1, 2, 3). Using property (), we give a short proof for a recent deviation inequality due to Talagrand. In a third section, we also recover known concentration results for Gaussian measures using our approach.     

Some uncertainty inequalities

We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.     

Some functional inequalities

LetM, N, O be open subsets of ? ⁿ and letF:M×N→O,f:O→?,g: M→?,h: N→? be functions, satisfying the functional inequality $$\forall (x,y) \in M \times N:f[F(x,y)] \leqslant g(x) + h(y).$$ IfF belongs to a certain extensive class of functions, we prove in this note, thatf is bounded above on every compact subset of ? ⁿ , wheneverh is bounded above on a Lebesgue-measurable set of positive Lebesgue-measure, contained inN (no assumptions aboutg are needed). Moreover we give applications of this theorem to generalized convex and subadditive functions.     

Linear inequalities concerning partitions into distinct parts

The Ramanujan Journal - Linear inequalities involving Euler’s partition function p(n) have been the subject of recent studies. In this article, we consider the partition function Q(n)...     

Some inequalities for the Bell numbers

In this paper, we present derivatives of the generating functions for the Bell numbers by induction and by the Faà di Bruno formula, recover an explicit formula in terms of the Stirling numbers of the second kind, find the (logarithmically) absolute and complete monotonicity of the generating functions, and construct some inequalities for the Bell numbers. From these inequalities, we derive the logarithmic convexity of the sequence of the Bell numbers.     

Some Stieltjes integral inequalities

In this paper some integral inequalities involving Lebesgue-Stieltjes integrals have been obtained which generalize, in particular, Bellman-Gronwall's inequality for Stieltjes integrals. Such inequalities will play an important role in the study of stability of impulsively perturbed systems.