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

a combinatorial identity involving gamma function and pochhammer symbol in this paper we establish an identity that relates pochhammer symbol and the ratio of gamma functions the identity is derived using the mellin series representation for the solution of a general algebraic equation

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

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

singularly perturbed ordinary differential equations abstract we show the existence of periodic solutions for a couple of ordinary differential equations depending on a parameter e when e presents a small parameter for the majorest derivative of the first equation the key idea is to connect a coincidence degree theory with a method from a previous paper by the author j differential equations 89 1991 203 223 to avoid the use of a hard implicit function theorem of nash moser

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

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

asymptotic formulas for the triple gamma function γ3 by means of its integral representation abstract there is an abundant literature on inequalities for the gamma function and its various related functions as well as their approximations only very recently several authors began to investigate various inequalities for the double gamma function 2 and its approximation here in this sequel to some of these recent works we aim at presenting an integral representation of the triple gamma function 3 which is then used to derive an asymptotic formula for 3 as a by product of the results presented here integral representations and asymptotic formulas for the gamma function and the double gamma function 2 are also given the methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple gamma functions n n 4

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

critical behavior of long linear k mers on honeycomb lattices abstract monte carlo mc simulations finite size scaling and theoretical analysis have been carried out to study the critical behavior of long linear particles of length k k mers on honeycomb lattices a nematic phase characterized by a big domain of parallel k mers is separated from the isotropic state by a continuous transition occurring at a finite density c our study allowed 1 to determine the minimum value of k k min which allows the formation of the nematic phase being k min 11 2 to predict the dependence of c on k being c k k 1 and 3 to obtain the critical exponents which indicate that the transition belongs to the 2d three state potts universality class

asymptotic behavior of the isotropic nematic and nematic columnar phase boundaries for the system of hard rectangles on a square lattice a system of hard rectangles of size m mk on a square lattice undergoes three entropy driven phase transitions with increasing density for large enough aspect ratio k first from a low density isotropic to an intermediate density nematic phase second from the nematic to a columnar phase and third from the columnar to a high density sublattice phase in this paper we show from extensive monte carlo simulations of systems with m 1 2 and 3 that the transition density for the isotropic nematic transition is a 1 k when k 1 where a 1 is independent of m we estimate a 1 4 80 0 05 within a bethe approximation and virial expansion truncated at the second virial coefficient we obtain a 1 2 the critical density for the nematic columnar transition when m 2 is numerically shown to tend to a value less than the full packing density as k 1 when k we find that the critical binder cumulant for this transition is nonuniversal and decreases as k 1 for k 1 however the transition is shown to be in the ising universality class

new fourier transform containing a pair of complex euler γ functions with a monomial mathematical and physical applications one of the goals of this paper is to propose an elementary method to find a general formula for a new fourier transform containing a pair of complex euler functions with a monomial sm in terms of gauss s hypergeometric functions 2f1 some mathematical results have been obtained and discussed using this transform furthermore physical applications in the phase space quantum mechanics requiring the expectation values of position and momentum operators for a quantum system endowed with position dependent effective mass are presented

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

some completely monotonic functions involving polygamma functions and an application by using the first binet s formula the strictly completely monotonic properties of functions involving the psi and polygamma functions are obtained as direct consequences two inequalities are proved as an application the best lower and upper bounds of the nth harmonic number are established

remarks on some completely monotonic functions applying the euler maclaurin summation formula we obtain upper and lower polynomial bounds for the function xex 1 x 0 with coefficients the bernoulli numbers bk this enables us to give simpler proofs of some results of h alzer and f qi et al concerning complete monotonicity of certain functions involving the gamma function x the psi function x and the polygamma functions n x n 1 2

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

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

singular perturbed problems in ordinary differential equations abstract here we solve singularly perturbed periodic problems in ordinary differential equations when singular perturbations contain two orders higher derivatives than unperturbed equations also we apply our method to the study of singularly perturbed variational problems

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

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

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

single polygon counting on cayley tree of order 3 we showed that one form of generalized catalan numbers is the solution to the problem of finding different connected component with finite vertices containing a fixed root for the semi infinite cayley tree of order 3 we give the formula for the full graph cayley tree of order 3 which is derived from the generalized catalan numbers using ratios of gamma functions two upper bounds are given for problem defined on semi infinite cayley tree of order 3 as well as the full graph

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

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

a new method for the existence of solutions of nonlinear differential equations many problems is analysis reduce to solving an equation au f where a is an operator on a space into another space u is the unknown and f is a given element the existence of a local solution near a trivial solution u0 of au 0 where 0 is near f is usually solved by the classical implicit function theorem as a matter of fact the picard iteration procedure is used unfortunately this procedure cannot be applied in some problems of partial differential equations these cases occur if the loss of derivatives appears 6 7 31 to get around this difficulty a more sophisticated iteration procedure is needed moser 1 has found such a method which enables one to handle a loss of derivatives by using a rapidly converging iteration scheme the nash moser technique is nicely described in the papers s lo we prove in this paper several theorems on the existence of solutions with the application to several types of specific problems the main interest is that in many cases these results can be used in situations which up to now have required the nash moser implicit function theorem yet they have classical proofs we also apply abstract results to show the existence of local invariant manifolds for singularly perturbed ordinary differential equations and we give a simple proof of a theorem of j moser for invariant tori l the plan of our paper is as follows in section 2 we present proofs of several existence theorems in an abstract setting in a scale of hilbert spaces the hypotheses are those commonly found in problems which have up to now required the application of the nash moser technique however the results have classical proofs based on the degree of mappings s we shall give in the proof of theorem 2 8 a new iteration procedure which retains supergeometric convergence properties we require obligations only for the linearization of our equation this is the main advance of this theorem section 3 of the paper presents applications of the theorems of section 2 there are demonstrated proofs of the above mentioned theorems of

a complete monotonicity property of the gamma function abstract a logarithmically completely monotonic function is completely monotonic the function 1 ln x 1 x ln x 1 is strictly completely monotonic on 0 the function x 1 x x is strictly logarithmically completely monotonic on 0

hard rigid rods on a bethe like lattice we study a system of long rigid rods of fixed length k with only excluded volume interaction we show that contrary to the general expectation the self consistent field equations of the bethe approximation do not give the exact solution of the problem on the bethe lattice in this case we construct a new lattice called the random locally treelike layered lattice which allows a dense packing of rods and we show that the bethe self consistent equations are exact for this lattice for a four coordinated lattice k mers with k 4 undergo a continuous isotropic nematic phase transition for even coordination number q 6 the transition exists only for k k min q and is discontinuous

a classical theory of hard squares a simple phenomenological theory of the hard square lattice gas is obtained by analyzing a low order corner transfer matrix variational approximation the free energy is of landau type and expressions are obtained for the order parameter and densities in this approximation the model exhibits a critical point atz c 4 3 2 3 9 with critical exponents given by the classical values 0disc 1 2 1 3

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

percolation clusters as generators for orientation ordering needles at different orientations are placed in an i i d manner at points of a poisson point process on mathbb r 2 of density lambda needles at the same direction have the same length while needles at different directions maybe of different lengths we study the geometry of a finite cluster when needles have only two possible orientations and when needles have only three possible orientations in both these cases the asymptotic shape of the finite cluster as lambda rightarrow infty is shown to consists of needles only in two directions in the two orientations case the shape does not depend on the orientation but just on the i i d structure of the orientations while in the three orientations case the shape depend on all the parameters i e the i i d structure of the orientations the lengths and the orientations of the needles in both these cases we obtain a totally ordered phase where all except one needle are bunched together with the exceptional needle binding them together

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

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

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

phase transitions in a system of hard rectangles on the square lattice the phase diagram of a system of monodispersed hard rectangles of size m mk on a square lattice is numerically determined for m 2 3 and aspect ratio k 1 2 7 we show the existence of a disordered phase a nematic phase with orientational order a columnar phase with orientational and partial translational order and a solidlike phase with sublattice order but no orientational order the asymptotic behavior of the phase boundaries for large k is determined using a combination of entropic arguments and a bethe approximation this allows us to generalize the phase diagram to larger m and k showing that for k 7 the system undergoes three entropy driven phase transitions with increasing density the nature of the different phase transitions is established and the critical exponents for the continuous transitions are determined using finite size scaling

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

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

some two sided inequalities for multiple gamma functions and related results there is an abundant literature on inequalities for the euler s gamma function c and its various related functions yet only very recently several authors began to study inequalities for the barnes double gamma function c 2 here in this paper we aim at presenting several two sided inequalities for the multiple gamma functions c n n 2 3 4 5 in our investigation of these two sided inequalities for the multiple gamma functions c n n 2 3 4 5 we employ and extend a method based upon taylor s formula and express email protected n 1 x as series involving the zeta functions we also give a more convenient explicit form of the multiple gamma functions c n email protected n n being the set of positive integers the main two sided inequalities for the multiple gamma functions c n n 2 3 4 5 which we have presented in this paper are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple gamma functions c n when email protected 6

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

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 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

certain summation formulas involving harmonic numbers and generalized harmonic numbers abstract harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science various branches of number theory elementary particle physics and theoretical physics here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function 5 f 4 1

on completely monotone of an arbitrary real parameter function involving the gamma function abstract in this article we present some completely monotone properties for an arbitrary real parameter function involving the gamma function as an application of these results some meaningful and useful inequalities are obtained