a class of logarithmically completely monotonic functions and the best bounds in the second kershaw s double inequality in the article the sufficient and necessary conditions such that a class of functions which involve the psi function j and the ratio c x t c x s are logarithmically completely monotonic are established the best bounds for the ratio c x t c x s are given and some comparisons with known results are carried out where s and t are two real numbers and x min s t

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

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

the knuth bendix completion procedure and thue systems the knuth bendix completion procedure for term rewriting systems in many cases provides a decision procedure for equational theories and has been found to have many applications in various areas we discuss the application of the knuth bendix procedure to thue systems we use the notion of a reduced thue system and show that for every church rosser thue system there is a unique reduced church rosser thue system equivalent to it furthermore the knuth bendix completion procedure when applied to a thue system t always produces the finite reduced church rosser thue system equivalent to t whenever such a system exists similar results can also be proved for almost confluent thue systems using properties of reduced church rosser systems we develop conditions under which a class of special thue systems have equivalent finite church rosser systems in addition we show that the completion procedure always terminates on finite parenthesized thue systems from which the termination of the completion procedur

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

representations of error terms in jensen s and some related inequalities with applications abstract we give integral representations of error terms in the classical jensen s inequality as well as in four related inequalities starting from these results it is possible to derive various lower and upper asymptotic expansions for solutions of the functional equation f x 1 f x g x in particular we present several results concerning the gamma function

complete characterizations of the gamma function abstract in this paper we obtain the complete monotonicities regarding both the functions x and x defined by x 1 2 x x e x e x 12 x 2 x x e x 1 x as direct consequences x is strictly increasing and strictly concave on 0 and x is strictly decreasing and strictly convex on 0 furthermore we show that x consists the properties lim x 0 x 0 and lim x x 1 as well as x holds the properties lim x 0 x and lim x x 0

some properties of a class of functions related to completely monotonic functions in this article we present several properties of the composition of functions which are related to the completely monotonic and absolutely monotonic functions relevant connections of the results derived in this article with those in earlier investigations are also indicated

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

monadic thue systems abstract certain infinite thue systems over a finite alphabet are studied in particular systems s e such that for each a e the set u u a s is a context freelanguage the syntactic structure of sets of ancestors and sets of descendants is considered as well as that of unions of congruence classes taken over infinite context free languages or regular sets the common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable even for finite systems the word problem for confluent systems of this type is shown to be tractable the question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation

the undecidability of the preperfectness of thue systems abstract since the word problem for thue systems was proved to be undecidable by post and markov in the 1940 s there has been some interest in developing conditions under which the word problem is decidable at present it is possible to decide whether a finite thue system is almost confluent or church rosser but testing for preperfectness has been an open problem for years here we prove this to be undecidable

optimum experimental design by shape optimization of specimens in linear elasticity the identification of lame parameters in linear elasticity is considered an optimum experimental design problem is formulated which aims at minimizing the size of an associated confidence ellipso

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

logarithmic convexity and inequalities for the gamma function abstract we propose a method based on logarithmic convexity for producing sharp bounds for the ratio x x as an application we present an inequality that sharpens and generalizes inequalities due to gautschi chu boyd lazarevic lupas and kershaw

some families of mathieu a series and alternating mathieu a series the main purpose of this paper is to present a number of potentially useful integral representations for the familiar mathieu a series as well as for its alternating version these results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the mathieu and alternating mathieu a series relationships of the mathieu a series with the riemann zeta function and the dirichlet eta function are also considered such special functions as the classical bessel function j n z and the confluent hypergeometric functions 0f 1 and 1f 2 are characterized by means of certain fredholm type integral equations of the first kind which are associated with some of these mathieu type series several integrals containing mathieu type series are also evaluated finally some closely related new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper

minimax lagrangian approach to the differentiability of nonlinear pde constrained shape functions without saddle point assumption the object of this paper is the computation of the domain or boundary expression of a state constrained shape function without explicitly resorting to the material derivative our main theorem bypasses the restrictive saddle point assumption in existing differentiability theorems of minimax by introducing the notion of the averaged adjoint state thus extending the use of minimax theorems to some classes of nonlinear state equations as an illustration the theorem is applied to a shape function that depends on a quasi linear transmission problem using a gagliardo penalization the existence of optimal shapes is established

some decision problems about controlled rewriting systems abstract a description of deterministic context free languages by some kind of finitely generated right congruences the sets of relations generating such right congruences are called controlled rewriting systems or c systems has been given previously in the first part of this paper we study several decision problems on c systems namely the confluence problem the equivalence problem the refinement problem the class equivalence problem and the class inclusion problem in the second part of the paper we deduce from the main theorem given previously a characterisation of dcfls by means of finitely generated congruences the sets of relations generating such congruences are now ordinary finite semi thue systems we study three decision problems on finite semi thue systems namely the class equivalence problem the word problem for the syntactic congruence of one class and the partial confluence problem all the problems are investigated in several classes of c systems or semi thue systems for every class we give an answer which may be yes the problem is decidable no the problem is undecidable or eq which means that the problem is recursively equivalent to the equivalence problem for dpda

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

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

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

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

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

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

continued fraction estimates for the psi function we present continued fraction estimates for the psi function as a consequence we educe inequality for the euler mascheroni constant which improves a result of detemple

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

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

efficient pde constrained shape optimization based on steklov poincare type metrics recent progress in pde constrained optimization on shape manifolds is based on the hadamard form of shape derivatives i e in the form of integrals at the boundary of the shape under investigation as well as on intrinsic shape metrics from a numerical point of view domain integral forms of shape derivatives seem promising which instead require an outer metric on the domain surrounding the shape boundary this paper tries to harmonize both points of view by employing a steklov poincare type intrinsic metric which is derived from an outer metric based on this metric efficient shape optimization algorithms are proposed which also reduce the analytical labor involved in the derivation of shape derivatives

a comparison theorem for two divided differences and applications to special functions abstract in this paper we present a general comparison theorem for two divided differences of a three times differentiable function this gives a unified treatment for logarithmically complete monotonicity monotonicity and inequalities involving some special functions including gamma psi and polygamma functions as their consequences we not only refine and generalize some important results but also present simple and interesting alternative proofs of certain earlier results

completely monotone solutions of the mode coupling theory for mixtures we establish that a mode coupling approximation for the dynamics of multi component systems obeying smoluchowski dynamics preserves a subtle yet fundamental property the partial density correlation functions are considered as matrices completely monotone i e they can exactly be written as superpositions of decaying exponentials only this statement holds no matter what further approximations are needed to calculate the theory s coupling parameters the long time limit of these functions fulfills a maximum property and an iteration scheme for its numerical determination is given we also show the existence of a unique solution to the equations of motion for which power series both for short times and small frequencies exist the latter except at special points where ergodic to nonergodic transitions occur these transitions are bifurcations that are proven to be of the cuspoid family

on the equivalence problem for regular thue systems abstract a decision procedure is presented for the equivalence problem for regular almost confluent thue systems on the other hand the equivalence problem for regular preperfect systems is shown to be undecidable

logarithmically completely monotonic functions relating to the gamma function abstract in this paper the logarithmically complete monotonicity of the function e x x x x in 0 for r and 0 is considered and the corresponding result by g d anderson r w barnard k c richards m k vamanamurthy and m vuorinen is generalized as applications of these results some inequalities between identric mean and ratio of two gamma functions by j d keckic and p m vasic are extended

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

some mathieu type series associated with the fox wright function closed form integral expressions are derived here for a family of convergent mathieu type series and its alternating variant when their terms contain the fox wright p j q function some two sided exponential bounding inequalities are then obtained for a class of fox wright p j q functions thereby generalizing certain results of luke finally by means of the integral expressions obtained here a number of two sided exponential bounding inequalities are given for the aforementioned series

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

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

a class of logarithmically completely monotonic functions associated with the gamma function in this article a necessary condition several sufficient conditions and a sufficient and necessary condition for a class of functions involving the gamma function to be logarithmically completely monotonic are established and then some known results are extended

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

shape sensitivity analysis via min max differentiability the object of this paper is twofold we introduce a new theorem on the differentiability of a min max with respect to a parameter and we show how such a theorem can be applied to compute the material derivative in shape sensitivity analysis problems we consider the min max of a functional which is parametrized by t we show that under appropriate conditions the derivative of the min max with respect to t is the min max with respect to the points solution of the min max problem of the derivative of the original functional with respect to t to illustrate the use of this theorem we apply it to the control of an elliptic equation with a nondifferentiable observation and to shape design problems

some properties of completely monotonic sequences and related interpolation in this article we present some properties of completely monotonic sequences based on our result related interpolation problems are then solved

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

complete monotonicity of some functions involving polygamma functions in the present paper we establish necessary and sufficient conditions for the functions x a j i x b and a j i x b x j i 1 x b respectively to be monotonic and completely monotonic on 0 where i n a 0 and b 0 are scalars and j i x are polygamma functions

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

improved asymptotic formulas for the gamma function the aim of this paper is to improve the ramanujan formula for approximation the gamma function a fast asymptotic series is constructed

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

monotonicity properties of functions related to the psi function abstract in this paper the monotonicity properties of functions related to the psi function are obtained a double inequality for the euler mascheroni constant is established relevant connections of the results presented here with those derived in earlier works are also pointed out

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

shape derivatives for the compressible navier stokes equations in variational form shape optimization based on surface gradients and the hadamard form is considered for a compressible viscous fluid special attention is given to the difference between the function composition approach involving local shape derivatives and an alternate methodology based on the weak form of the state equation the resulting gradient expressions are found to be equal only if the existence of a strong form solution is assumed surface shape derivatives based on both formulations are implemented within a discontinuous galerkin flow solver of variable order the gradient expression stemming from the variational approach is found to give superior accuracy when compared to finite differences

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