uniform boundary harnack principle and generalized triangle property let d be a bounded open subset in rd d 2 and let g denote the green function for d with respect to 2 0 2 d if 2 assume that d satisfies the interior corkscrew condition if 2 i e if g is the classical green function on d assume more restrictively that d is a uniform domain let g g y0 1 for some y0 d based on the uniform boundary harnack principle it is shown that g has the generalized triangle property which states that g z y g z cg x y g x when d z x d z y an intermediate step is the approximation g x y x y dg x g y g a 2 where a is an arbitrary point in a certain set b x y this is discussed in a general setting where d is a dense open subset of a compact metric space satisfying the interior corkscrew condition and g is a quasi symmetric positive numerical function on d d which has locally polynomial decay and satisfies harnack s inequality under these assumptions the uniform boundary harnack principle the approximation for g and the generalized triangle property turn out to be equivalent

further inequalities for the gamma function for x 0 and k 0 we present a method which permits us to obtain inequalities of the type k a x 1 r k x r k 1 k x 1 with the usual notation for the gamma function where a and are independent of k some examples are also given which improve well known inequalities finally we are also able to show in some cases that the values a and in the inequalities that we obtain cannot be improved

on gamma function inequalities on montre que certaines fonctions comportant les quotients de fonctions gamma sont completement monotones on en deduit des inegalites on etablit aussi la divisibilite infinie de plusieurs distributions de probabilite dont les transformees de laplace comportent des quotients de fonctions gamma

kolmogorov landau inequalities for monotone functions where llfll ess su if t kolmogorov 2 suceeded in finding the best possible constants in 1 where the norm is as in 2 except for isolated results no new progress was made until schoenberg and cavaretta 3 found the best possible constants for llfll su if f we propose to prove inequalities about various clases of monotone functions which yield as special cases pointwise inequalities of the form 1 for example

convexity properties of harmonic measures it is shown that any convex combination of harmonic measures where u1 uk are relatively compact open neighborhoods of a given point d2 can be approximated by a sequence of harmonic measures such that each wn is an open neighborhood of x in u1uk this answers a question raised in connection with jensen measures moreover it implies that for every green domain x containing x the extremal representing measures for x with respect to the convex cone of potentials on x these measures are obtained by balayage of the dirac measure at x on borel subsets of x are dense in the compact convex set of all representing measures this is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner these results which are presented simultaneously for the classical potential theory and for the theory of riesz potentials can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property the methods developed for this purpose continuous balayage on increasing families of compact sets approximation using scattered sets with small capacity finally lead to answers even in a very general potential theoretic setting covering a wide class of second order partial differential operators uniformly elliptic or in divergence form or sums of squares of vector fields satisfying hormander s condition for example sub laplacians on stratified lie algebras

some new asymptotic approximations of the gamma function based on nemes formula ramanujan s formula and burnside s formula in this paper we construct some new approximations of the gamma function based on nemes formula ramanujan s formula and burnside s formula using these approximations some inequalities are established finally for demonstrating the superiority of our new approximation over mortici s formula and other classical ones some numerical computations are also given

ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

completely monotonic degrees for a difference between the logarithmic and psi functions abstract in the paper the authors firstly present a concise proof for complete monotonicity of a function involving a difference between the logarithmic and psi functions secondly compute completely monotonic degree of the above mentioned function and finally pose several conjectures on completely monotonic degrees of remainders and their derivatives for the asymptotic formula of the logarithm of the classical euler gamma function

absolute continuity and singularity of probability measures induced by a purely discontinuous girsanov transform of a stable process in this paper we study mutual absolute continuity and singularity of probability measures on the path space which are induced by an isotropic stable levy process and the purely discontinuous girsanov transform of this process we also look at the problem of finiteness of the relative entropy of these measures an important tool in the paper is the question under which circumstances the a s finiteness of an additive functional at infinity implies the finiteness of its expected value 2010 mathematics subject classification primary 60j55 secondary 60g52 60j45 60h10

an optimal algorithm for constructing the weighted voronoi diagram in the plane let s denote a set of n points in the plane such that each point p has assigned a positive weight w p which expresses its capability to influence its neighbourhood in this sense the weighted distance of an arbitrary point x from p is given by de x p w p where de denotes the euclidean distance function the weighted voronoi diagram for s is a subdivision of the plane such that each point p in s is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of s an algorithm which constructs the weighted voronoi diagram for s in o n2 time is outlined in this paper the method is optimal as the diagram can consist of n2 faces edges and vertices

on stirling s formula remainder let the sequence r n be defined by n 2 n n e n e r n we establish new estimates for stirling s formula remainder r n this improves some known results let x be defined by the relation x 1 2 x e x e x 12 x we prove that x is completely monotonic on 0 this implies the result given by mortici who proved that x 1 x is completely monotonic on 0

sharp bounds for gamma function in terms of x x 1 the aim of this paper is to present new approximations for the gamma function in terms of x x 1 the monotonicity and convexity of some involved functions are studied as applications sharp bounds are presented it is also solved an open problem posed by mortici 2013

inequalities for the harmonic numbers we present various inequalities for the harmonic numbers defined by h n 1 1 2 ldots 1 n n in bf n one of our results states that we have for all integers n 2 alpha frac log log n gamma n 2 leq h n 1 n h n 1 1 n 1 beta frac log log n gamma n 2 with the best possible constant factors alpha frac 6 sqrt 6 2 sqrt 3 396 3 log log 2 gamma 0 0140 ldots quad mbox and quad beta 1 here denotes euler s constant

an inequality for ratios of gamma functions abstract let x denote euler s gamma function the following inequality is proved for y 0 and x 1 we have x y 1 y 1 1 x x y 2 y 1 1 x 1 x y x y 1 the inequality is reversed if 0 x 1 this resolves an open problem of guo and qi b n guo f qi inequalities and monotonicity for the ratio of gamma functions taiwanese j math 7 2003 239 247

supplements to a class of logarithmically completely monotonic functions associated with the gamma function abstract in this article the authors present a necessary condition a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic as a consequence of these results supplements to the recent investigation by chen and qi c p chen f qi logarithmically completely monotonic functions relating to the gamma function j math anal appl 321 2006 405 411 are provided and a new keckic vasic type inequality is concluded

asymptotic estimates for stieltjes constants a probabilistic approach let n n 0 be the sequence of stieltjes constants appearing in the laurent expansion of the riemann zeta function we obtain explicit upper bounds for n whose order of magnitude is as n tends to

two sided heat kernel estimates for censored stable like processes in this paper we study the precise behavior of the transition density func tions of censored resurrected stable like processes in c 1 1 open sets in r d where d 1 and 1 2 we first show that the semigroup of the censored stable like process in any bounded lipschitz open set is intrinsically ultracontractive we then establish sharp two sided estimates for the transition density functions of a large class of censored stable like processes in c 1 1 open sets we further obtain sharp two sided estimates for the green functions of these censored stable like processes in bounded c 1 1 open sets

mystery of point charges we discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in rn this question goes back to j c maxwell 10 and m morse

on the hausdorff dimension of harmonic measure in higher dimension assume a i n e a domain in f n where e is a compact set denote o a a x the harmonic measure for a of a evaluated at x r d according to 0ksendal s theorem o o e o a x is singular with respect to d dimensional lebesgue measure for d 2 and general domains this result seemed to be so far the only known localization property recently for d 2 it has been shown by p jones and t wolff j w that s o9 has dimension at most 1 this result completes previous work due to n g makarov and l carleson see i m and c2 let g be the green s function of a r with pole at some point of c r o9 in both the carleson and jones wolff arguments the integral g g nn log nn as boundary

some properties of the divided difference of psi and polygamma functions abstract let n 1 n 1 n for n 0 where n stands for the psi and polygamma functions for p q r and min p q let d x p x q n 1 n x be the divided difference of the functions n 1 for x in this paper we establish the necessary and sufficient conditions for the function n x n 1 x 2 n x n 2 x to be completely monotonic on in particular we find that the function n 1 2 n n 2 is strictly decreasing from 0 onto n n 1 n 1 n 2 these not only generalize and strengthen some known results but also yield many new and interesting ones

quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions we show that ii a set e is dispersely decomposed into subsets then the capacity of e is comparable to the summation of tile capacities of the subsets from this fact it is derived that the lebesgue measure of a certain expanded set is estimated by the ca

sinr diagrams convexity and its applications in wireless networks the rules governing the availability and quality of connections in a wireless network are described by physical models such as the signal to interference noise ratio sinr model for a collection of simultaneously transmitting stations in the plane it is possible to identify a reception zone for each station consisting of the points where its transmission is received correctly the resulting sinr diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard sinr diagrams appear to be fundamental to understanding the behavior of wireless networks and may play a key role in the development of suitable algorithms for such networks analogous perhaps to the role played by voronoi diagrams in the study of proximity queries and related issues in computational geometry so far however the properties of sinr diagrams have not been studied systematically and most algorithmic studies in wireless networking rely on simplified graph based models such as the unit disk graph udg model which conveniently abstract away interference related complications and make it easier to handle algorithmic issues but consequently fail to capture accurately some important aspects of wireless networks this article focuses on obtaining some basic understanding of sinr diagrams their properties and their usability in algorithmic applications specifically we have shown that assuming uniform power transmissions the reception zones are convex and relatively well rounded these results are then used to develop an efficient approximation algorithm for a fundamental point location problem in wireless networks

hypergeometric functions as generalized stieltjes transforms abstract in this paper we apply generalized stieltjes transform representation to study the generalized hypergeometric function among the results thus proved are new integral representations inequalities properties of the pade table and the properties of the generalized hypergeometric function as a conformal map

a class of strongly completely monotonic functions related to gamma function abstract a function f 0 r is called strongly completely monotonic if it has derivatives of all orders and 1 n x n 1 f n x is nonnegative and decreasing on 0 for all n 0 in this paper we prove the function g n x ln x 1 2 x ln x x 1 2 ln 2 k 1 n 1 2 1 2 k b 2 k 2 k 2 k 1 x 2 k 1 is strongly completely monotonic on 0 using the same technique we give an alternative proof of a known result moreover two conjectures are proposed

some gamma function inequalities a class of completely monotonic functions are presented involving the gamma function as well as the derivative of the psi function as a consequence new upper and lower bounds for the ratio f x 1 f x s are obtained and compared with related bounds given in part by j d keckic and p m vasic our results are further applied to obtain functions which are laplace transforms of infinitely divisible probability measures

sharp bounds of the landau constants the aim of this paper is to establish new bounds of the landau constants 1 introduction and motivation the landau constants de ned for all positive integers n by gn 1 1 2 2 1 3 2 4 2 1 3 5 2 4 6 2 1 3 2n 1 2 4 2n 2 play an important role in some extremal problems in complex analysis and in the theory of fourier series more precisely in 1913 landau 6 proved that gn is the maximum of the expression j pn k 1 akj with respect to all functions of the form f z p1 k 0 akz k which is analytic in the unit disk and satis es jf z j 1 for every jzj 1 in consequence the problem of approximation of the landau constants have attracted the attention of many authors in particular landau himself studied the asymptotic behaviour of gn and showed that gn 1 lnn then watson 7 established the following asymptotic formula gn c0 1 ln n 1 1 4 n 1 o 1 n2 n 1 here and in what follows c0 1 4 ln 2 1 06627 and 0 577215 is the euler mascheroni constant this problem of approximation of the landau constants was continued in the works of brutman 3 and falaleev 5 who proved that for every non negative integer n 1 1 ln n 1 gn 1 0663 1 ln n 1 respective 1 0662 1 ln n 3 4 gn 1 0916 1 ln n 3 4 we improve the upper bound in the following way that also shows that the constant 3 4 is the best possible date november 18 2009 1991 mathematics subject classi cation primary 41a60 secondary 26d15

exponential gamma and polygamma functions simple proofs of classical and new inequalities abstract some inequalities for the exponential gamma and polygamma functions are established by means of the mean value theorem of differential calculus in particular the cases of the ratio of gamma and polygamma functions are considered in some cases the results are new in some others classical results are recovered or extended

gurland s ratio for the gamma function we consider the ratio t x y x y 2 x y 2 and its properties related to convexity logarithmic convexity schur convexity and complete monotonicity several new bounds and asymptotic expansions for t are derived sharp bounds for the function x x 1 e x are presented as well as bounds for the trigamma function the results are applied to a problem related to the volume of the unit ball in r n and also to the problem of finding the inverse of the function x t 1 x 3 x which is of importance in applied statistics

inequalities for the median of the gamma distribution we provide several new inequalities involving l n the median of the gamma distribution of order n 1 with parameter 1 among others we present sharp upper and lower bounds for the arithmetic mean of l 1 l 2 l n for all integers n 1 we have an 76 n2 8405h nn email protected email protected k bn 76 n2 8405h nn with the best possible constants email protected k 1 l k k 23 8405k email protected email protected 1 683405 0 0080 here h n denotes the nth harmonic number

on a ramanujan equation connected with the median of the gamma distribution in this paper we consider the sequence n n 0 solving the ramanujan equation e n 2 n k 0 n k k n n n n 1 n 0 1 the three main achievements are the following we introduce a continuous time extension 0 t of on and show its close connections with the medians an of the n 1 1 distributions and the charlier polynomials we give upper and lower bounds for both 0 t and an in particular for on which are sharper than other known estimates finally we show and at the same time complete two conjectures by chen and rubin referring to the sequence of medians n n 1

on ramanujan s q function abstract this study provides a detailed analysis of a function which knuth discovered to play a central role in the analysis of hashing with linear probing the function named after knuth q n is related to several of ramanujan s investigations it surfaces in the analysis of a variety of algorithms and discrete probability problems including hashing the birthday paradox random mapping statistics the rho method for integer factorization union find algorithms optimum caching and the study of memory conflicts a process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the q n function in this way tight bounds are derived they answer a question of knuth the art of computer programming vol 1 1968 ex 1 2 11 3 13 itself a rephrasing of earlier questions of ramanujan in 1911 1913

stochastic complexities of gaussian mixtures in variational bayesian approximation bayesian learning has been widely used and proved to be effective in many data modeling problems however computations involved in it require huge costs and generally cannot be performed exactly the variational bayesian approach proposed as an approximation of bayesian learning has provided computational tractability and good generalization performance in many applications the properties and capabilities of variational bayesian learning itself have not been clarified yet it is still unknown how good approximation the variational bayesian approach can achieve in this paper we discuss variational bayesian learning of gaussian mixture models and derive upper and lower bounds of variational stochastic complexities the variational stochastic complexity which corresponds to the minimum variational free energy and a lower bound of the bayesian evidence not only becomes important in addressing the model selection problem but also enables us to discuss the accuracy of the variational bayesian approach as an approximation of true bayesian learning

monotonicity of some functions involving the gamma and psi functions let l x x x t x s x 8 t 1 where x is euler s gamma function we determine conditions for the numbers s t so that the function x x s x t x t s 1 l x is strongly completely monotonic on 0 oo through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation we also give several other examples of strongly completely monotonic functions defined in terms of and functions some limiting and particular cases are also considered

green function estimate for censored stable processes sharp two sided estimates for green functions of censored stable process y in a bounded c 1 1 open set d are obtained where 1 2 it is shown that the martin boundary and minimal martin boundary of y can all be identified with the euclidean boundary of d sharp two sided estimates for the martin kernel of y are also derived

inequalities for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 by gn k 0n1 16k 2 and ln 12 sin n t sin t dt respectively we establish sharp inequalities for gn and ln 2 in terms of the logarithmic derivative of the gamma function further we prove that the sequence gn is completely monotonic we provide best possible upper and lower bounds for the ratios gn 1 gn 1 gn and l n 1 2 l n 1 2 ln 2 and we present sharp bounds for ln 2 gn and ln 2 gn

the topology of wireless communication this article studies the topological properties of wireless communication maps and their usability in algorithmic design we consider the sinr model which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise to describe the behavior of a multistation network we use the convenient representation of a reception map which partitions the plane into reception zones one per station and the complementary region of the plane where no station can be heard sinr diagrams have been studied in avin et al 2009 for the specific case where all stations use the same power it was shown that the reception zones are convex hence connected and fat and this was used to devise an efficient algorithm for the fundamental problem of point location here we consider the more general and common case where transmission energies are arbitrary or nonuniform under that setting the reception zones are not necessarily convex or even connected this poses the algorithmic challenge of designing efficient point location techniques for the nonuniform setting as well as the theoretical challenge of understanding the geometry of sinr diagrams e g the maximal number of connected components they might have our key result exhibits a striking contrast between d and d 1 dimensional maps for a network embedded in d dimensional space specifically it is shown that whereas the d dimensional map might be highly fractured drawing the map in one dimension higher heals the zones which become connected in fact hyperbolically connected we also provide bounds for the fatness of reception zones subsequently we consider algorithmic applications and propose a new variant of approximate point location

estimates of generalized stieltjes constants with a quasi geometric rate of decay we approximate each generalized stieltjes constant n a by means of a finite sum involving bernoulli numbers such an approximation has a quasi geometric rate of convergence which improves as re a increases a more detailed analysis including numerical computations is carried out for the constants 1 and 1 1 the key point in the proof is a probabilistic representation of the aforementioned constants obtained as a consequence of a differential calculus concerning the gamma process

limits and inequalities associated with the euler mascheroni constant i we present several limits associated with the euler mascheroni constant ii let c 0 577215 be the euler mascheroni constant and let t n k 1 n1k lnn 12 124n and p n k 1 n22k 1 ln 4n we determine the best possible constants a b a and b such that the inequalities148 n a 3 c t n 148 n b 3and124 n a 2 1

the asymptotic series of the generalized stirling formula the aim of this paper is to construct some asymptotic expansions which produce increasingly accurate approximations of the gamma function

series associated to some expressions involving the volume of the unit ball and applications the aim of this paper is to construct asymptotic series associated to some expressions involving the volume of the n dimensional unit ball new refinements and sharpenings of some old and recent inequalities on the volume of the n dimensional unit ball are presented

the dirichlet problem for sublaplacians on nilpotent lie groups geometric criteria for regularity on etudie un critere de regularite geometrique pour un sous ensemble ouvert relativement compact u d un espace harmonique

continued fraction approximation for the gamma function based on the tri gamma function abstract in this paper we present a continued fraction product approximation for the gamma function via the tri gamma function this approximation is fast in comparison with the recently discovered asymptotic series we also establish the inequalities related to this approximation finally some numerical computations are provided for demonstrating the superiority of our approximation

an infinite class of completely monotonic functions involving the q gamma function abstract in this paper the q analogue of the stirling formula for the q gamma function is used to prove the complete monotonicity property for an infinite class of functions which are all closely related to the q gamma function and its logarithmic derivatives q polygamma functions as an application of this result the two sided inequalities for the q gamma and the q polygamma functions are established

champagne subdomains with unavoidable bubbles abstract a champagne subdomain of a connected open set u 0 in r d d 2 is obtained by omitting pairwise disjoint closed balls b x r x x x the bubbles where x is an infinite locally finite set in u the union a of these balls may be unavoidable that is brownian motion starting in u a and killed when leaving u may hit a almost surely or equivalently a may have harmonic measure 1 for u a recent publications by gardiner and ghergu d 3 and by pres d 2 give rather sharp answers to the question of how small such a set a may be when u is the unit ball in this paper using a totally different approach optimal results are obtained which hold also for arbitrary connected open sets u

inequalities and monotonicity properties for the gamma function we present a survey of some recent results on the ratio of two gamma functions and prove the following inequalities for the product 1 1 x 1 x 2 1 x 1 1 x 2 0 and x 1 x 2 1 finally we study the possible extension to k variables of the lower bound above

completely monotonic functions of positive order and asymptotic expansions of the logarithm of barnes double gamma function and euler s gamma function abstract we introduce completely monotonic functions of order r 0 and show that the remainders in asymptotic expansions of the logarithm of barnes double gamma function and euler s gamma function give rise to completely monotonic functions of any positive integer order

some properties of the gamma and psi functions with applications in this paper some monotoneity and concavity properties of the gamma beta and psi functions are obtained from which several asymptotically sharp inequalities follow applying these properties the authors improve some well known results for the volume n of the unit ball b n r n the surface area n 1 of the unit sphere s n 1 and some related constants

inequalities for the volume of the unit ball in rn abstract let n n 2 1 n 2 be the volume of the unit ball in rn we determine the best possible constants a b a b and such that the inequalities a n n 1 n 1 n b n n 1 n 1 n a 2 n 1 n n b 2 and 1 1 n 2 n n 1 n 1 1 1 n are valid for all integers n 1 our results refine and complement inequalities proved by g d anderson et al k h borgwardt and d a klain and g c rota

improved time censored reliability test plans for k out of n gamma systems abstract relevant prior knowledge supported by historical data and expert judgments is common in many manufacturing processes the classical viewpoint to the determination of acceptance sampling plans for k out of n gamma systems is generalized to those situations in which the available previous information is appreciable a beta distribution with a reduced parameter space is adopted to describe the stochastic behavior of the fraction of defective systems an integer nonlinear programming problem is stated to find the inspection scheme that minimizes the required number of components to test and keeps the average producer and consumer risks below specified thresholds first lower and upper bounds as well as an approximate solution are derived then an efficient step by step procedure which finds the best acceptance sampling plan in a finite number of iterations is provided the inclusion of prior information into the decision process provides substantial savings in the sample size required for demonstrating system reliability and also more accurate assessments of the actual producer and consumer risks a system of five water pumps for cooling a reactor is analyzed for illustrative and comparative purposes

estimates of the function and quotient by minc sathre we present some approximations of minc sathre quotient and some related lower and upper bounds then we establish some estimates of minc sathre function and the nth root of the volume of the unit n dimensional ball some applications and comparison with other results are given

a note on the volume of sections of bnp abstract let b p n denote the unit ball in l p n with p 1 we prove that vol n 1 h b p n vol n b p n n 1 n for any n 1 dimensional subspace h of r n this is a consequence of bounding the isotropy constant of b p n above by 1 12 and we show that one can replace 1 12 by a possibly smaller number for n 2

a turan type inequality for the gamma function abstract we prove that the following turan type inequality holds for euler s gamma function for all odd integers n 1 and real numbers x 0 we have n 1 x n 1 x n x 2 with the best possible constant min 1 5 x 2 x 2 x 0 6359

new integral representations for the fox wright functions and its applications abstract our aim in this paper is to derive several new integral representations for the fox wright functions in particular we give new laplace and stieltjes transforms for this special function under some restrictions on parameters from the positivity conditions on the weight in these representations we found sufficient conditions to be imposed on the parameters of the fox wright functions which allow us to conclude that it is completely monotonic as applications we derive a class of functions that are related to the fox h functions and are positive definite moreover we extended the luke s inequalities and we establish new turan type inequalities for the fox wright function finally by appealing to each of the luke s inequalities two sets of two sided bounding inequalities for the generalized mathieu s type series are proved

on ruijsenaars asymptotic expansion of the logarithm of the double gamma function abstract we study the remainder r n x in an asymptotic expansion due to s n m ruijsenaars for the logarithm of the double gamma function we show that for any even number n the function 1 n 2 1 r n x is completely monotonic on 0 this proves a recent conjecture of h l pedersen in addition we give an estimate for the remainder r n x for n even

some extensions of w gautschi s inequalities for the gamma function it has been shown by w gautschi that if 0 i xi s f x exp i s x 1 the following closer bounds are proved exp i s 4 x 12 f exp i s x s i f x s 2 and x 2 t x 2 s 4 these are compared with each other and with inequalities given by t erber and j d keckic and p m vasic

completely monotonic functions associated with the gamma function and its q analogues several functions involving the gamma function x and the q gamma function q x are proved to be completely monotonic some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions

unavoidable collections of balls for censored stable processes abstract we study avoidability of collections of balls in bounded c 1 1 opens sets for censored stable processes 1 2 the results are analogous to the ones obtained for brownian motion in gardiner and ghergu 2010 11 on the way we derive a wiener aikawa type criterion for minimal thinness with respect to the censored stable processes

some inequalities and monotonicity properties associated with the gamma and psi functions abstract in this paper we present new completely monotonic functions and bernstein functions relating to the gamma and psi functions as consequences we establish new upper and lower bounds for the gamma function in terms of function also we present some estimates for stirling s formula remainder

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

logarithmically completely monotonic functions and applications in this article we give a sufficient condition and a necessary and sufficient condition for a function class associated with the gamma function to be logarithmically completely monotonic some applications are also presented

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

censored stable processes we present several constructions of a censored stable process in an open set d rn i e a symmetric stable process which is not allowed to jump outside d we address the question of whether the process will approach the boundary of d in a finite time we give sharp conditions for such approach in terms of the stability index and the thickness of the boundary as a corollary new results are obtained concerning besov spaces on non smooth domains including the critical exponent case we also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary we derive a boundary harnack principle in c1 1 open sets