on the alternating series 1 12 13 14 abstract it is shown that the best constants a and b such that inequalities 1 2n a k n 1 1 k 1 1 k 1 2n b hold for every n 1 are a 1 1 log 2 2 1 258891 and b 1

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

low firing rates in a compartmental model neuron a nonlinear relationship is obtained between the membrane potential of a compartmental model neuron as a function of input stimulation this is used to show how shunting effects can lead to low output firing rates in associative memory networks

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

markowitz s mean variance portfolio selection with regime switching a continuous time model a continuous time version of the markowitz mean variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states the random regime switching is assumed to be independent of the underlying brownian motion this essentially renders the underlying market incomplete a markov chain modulated diffusion formulation is employed to model the problem using techniques of stochastic linear quadratic control mean variance efficient portfolios and efficient frontiers are derived explicitly in closed forms based on solutions of two systems of linear ordinary differential equations related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed all the results are markedly different from those for the case when there is no regime switching an interesting observation is however that if the interest rate is deterministic then the results exhibit rather unexpected similarity to their no regime switching counterparts even if the stock appreciation and volatility rates are markov modulated

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

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 monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions we present some complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions this extends some known results due to s l qiu and m vuorinen

improved bounds on lossless source coding and guessing moments via renyi measures this paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available these moments quantify the number of guesses required for correctly identifying the unknown object and similarly to arikan s bounds they are expressed in terms of the arimoto renyi conditional entropy although arikan s bounds are asymptotically tight the improvement of the bounds in this paper is significant in the non asymptotic regime relationships between moments of the optimal guessing function and the map error probability are also established characterizing the exact locus of their attainable values the bounds on optimal guessing moments serve to improve non asymptotic bounds on the cumulant generating function of the codeword lengths for fixed to variable optimal lossless source coding without prefix constraints non asymptotic bounds on the reliability function of discrete memoryless sources are derived as well relying on these techniques lower bounds on the cumulant generating function of the codeword lengths are derived by means of the smooth renyi entropy for source codes that allow decoding errors

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

on some properties of digamma and polygamma functions in this note we present some new and structural inequalities for digamma polygamma and inverse polygamma functions we also extend generalize and refine some known inequalities for these important functions

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

on leibniz series defined by convex functions abstract it is shown that for every 0 we have k n 1 1 k 1 k 1 2 n n for some strictly decreasing sequence n n 1 such that 1 2 n 1 2 1 1 2 n 1 1 hence with lim n n 1 2 this is only a particular case of more general new results on leibniz series defined by convex functions

monotonicity rules for the ratio of two laplace transforms with applications abstract let f and g be both continuous functions on 0 with g t 0 for t 0 and let f x l f g x l g be respectively the laplace transforms of f and g converging for x 0 we prove that if there is a t 0 such that f g is strictly increasing on 0 t and strictly decreasing on t then the ratio f g is decreasing on 0 if and only if h f g 0 lim x 0 f x g x g x f x 0 with lim x 0 f x g x lim t f t g t and lim x f x g x lim t 0 f t g t provided the indicated limits exist while h f g 0 0 there is at least one x 0 such that f g is increasing on 0 x and decreasing on x as applications a unified treatment for certain bounds of psi function is presented and some properties of the modified bessel functions of the second are established these show that the monotonicity rules in this paper may contribute to study for certain special functions because many special functions can be expressed as corresponding laplace transforms

numerical approximation for a superreplication problem under gamma constraints we study a superreplication problem of european options with gamma constraints in mathematical finance the initially unbounded control problem is set back to a problem involving a viscosity pde solution with a set of bounded controls then a numerical approach is introduced unconditionally stable with respect to the mesh steps a generalized finite difference scheme is used since basic finite differences cannot work in our case numerical tests illustrate the validity of our approach

a stochastic representation for mean curvature type geometric flows a smooth solution t t 0 t rd of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem in this control problem the controller tries to steer the state process into a given deterministic set tc with probability one the reachability set v t for the target problem is the set of all initial data x from which the state process xx t tc for some control process this representation is proved by studying the squared distance function to t for the codimension k mean curvature flow the state process is dx t 2 pdw t where w t is a d dimensional brownian motion and the control p is any projection matrix onto a d k dimensional plane smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given

on the numerical solution of nonlinear black scholes equations nonlinear black scholes equations have been increasingly attracting interest over the last two decades since they provide more accurate values by taking into account more realistic assumptions such as transaction costs risks from an unprotected portfolio large investor s preferences or illiquid markets which may have an impact on the stock price the volatility the drift and the option price itself in this paper we will focus on several models from the most relevant class of nonlinear black scholes equations for european and american options with a volatility depending on different factors such as the stock price the time the option price and its derivatives due to transaction costs we will analytically approach the option price by transforming the problem for a european call option into a convection diffusion equation with a nonlinear term and the free boundary problem for an american call option into a fully nonlinear nonlocal parabolic equation defined on a fixed domain following sevcovic s idea finally we will present the results of different numerical discretization schemes for european options for various volatility models including the leland model the barles and soner model and the risk adjusted pricing methodology model

convexity according to a pair of quasi arithmetic means and inequalities abstract we consider a class of generalized convex functions which are defined according to a pair of quasi arithmetic means and called m m convex functions and establish various fejer type inequalities for such a function class these inequalities not merely provide a natural and intrinsic characterization of the m m convex functions but actually offer a generalization and refinement of the most part of the concrete hermite hadamard and fejer type inequalities obtained in earlier studies for different kinds of convexity and fractional integrals applications to inequalities involving the gamma function and special means are also included

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 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

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

wendel s and gautschi s inequalities refinements extensions and a class of logarithmically completely monotonic functions abstract in the article sufficient and necessary conditions that a class of functions involving ratio of euler s gamma functions and originating from wendel s and gautschi s inequalities are logarithmically completely monotonic are presented from this wendel s gautschi s kershaw s laforgia s bustoz ismail s merkle s and elezovic giordano pecaric s inequalities are refined extended and sharpened and a double inequality on the divided differences of the psi and polygamma functions is deduced straightforwardly

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

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

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

superreplication under gamma constraints in a financial market consisting of a nonrisky asset and a risky one we study the minimal initial capital needed in order to superreplicate a given contingent claim under a gamma constraint this is a constraint on the unbounded variation part of the hedging portfolio we first consider the case in which the prices are given as general markov diffusion processes and prove a verification theorem which characterizes the superreplication cost as the unique solution of a quasi variational inequality in the context of the black scholes model i e when volatility is constant this theorem allows us to derive an explicit solution of the problem these results are based on a new dynamic programming principle for general stochastic target problems

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

sharp bounds for psi function in this paper we present the greatest value a 1 15 and the least value b 4 15 such that the double inequality l x a 0 where l x a 1 90 a 2 2 log x 2 x 3 a 1 3 45 a 2 90 a 2 2 log x 2 x 15 a 1 45 a and x is the classical psi function

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

simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension we introduce simplex free adaptive tree numerical methods for solving static and time dependent hamilton jacobi equations arising in level set problems in arbitrary dimension the data structure upon which our method is built in a generalized n dimensional binary tree but it does not require the complicated splitting of cubes into simplices aka generalized n dimensional triangles or hypertetrahedrons that current tree based methods require it has enough simplicity that minor variants of standard numerical hamiltonians developed for uniform grids can be applied yielding consistent monotone convergent schemes combined with the fast sweeping strategy the resulting tree based methods are highly efficient and accurate thus without changing more than a few lines of code when changing dimension we have obtained results for calculations in up to n 7 dimensions

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 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

integral estimates for convergent positive series abstract it is shown that for every 1 we have k n 1 1 k 1 1 n n 1 for some strictly decreasing sequence n n 1 such that 1 2 n 1 4 1 1 1 2 n 1 hence with lim n n 1 2 this is only a particular case of more general new results on series defined by convex functions

hedging of covered options with linear market impact and gamma constraint within a financial model with linear price impact we study the problem of hedging a covered european option under gamma constraint using stochastic target and partial differential equation smoothing techniques we prove that the super replication price is the viscosity solution of a fully non linear parabolic equation as a by product we show how e optimal strategies can be constructed finally a numerical resolution scheme is proposed

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

the dynamic programming equation for second order stochastic target problems motivated by applications in mathematical finance u cetin h m soner and n touzi options hedging for small investors under liquidity costs finance stoch to appear we continue our study of second order backward stochastic equations in this paper we derive the dynamic programming equation for a certain class of problems which we call the second order stochastic target problems in contrast with previous formulations of similar problems we restrict control processes to be continuous this new framework enables us to apply our results to a larger class of models also the resulting derivation is more transparent the main technical tool is the geometric dynamic programming principle in this context and it is proved by using the framework developed in h m soner and n touzi j eur math soc jems 8 2002 pp 201 236

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

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

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

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

optimal control versus stochastic target problems an equivalence result abstract within a general abstract framework we show that any optimal control problem in standard form can be translated into a stochastic target problem as defined in soner and touzi 2002 5 whenever the underlying filtered probability space admits a suitable martingale representation property this provides a unified way of treating these two classes of stochastic control problems as an illustration we show within a jump diffusion framework how the hamilton jacobi bellman equations associated to an optimal control problem in standard form can be easily retrieved from the partial differential equations associated to its stochastic target counterpart

bethe ansatz equations for quantum chains combining different representations of su 3 the general expression for the local matrix of a quantum chain with the site space in any representation of is obtained this is made by generalizing from the fundamental representation and imposing the fulfilment of the yang baxter equation with these operators and using a generalization of the nested bethe ansatz the bethe equations for a multistate quantum chain combining two arbitrary representations of su 3 are obtained

reciprocally convex functions we say that f is reciprocally convex if x f x is concave and x f 1 x is convex on 0 reciprocally convex functions generate a sequence of quasi arithmetic means with the first one between harmonic and arithmetic mean and others above the arithmetic mean we present several examples related to the gamma function and we show that if f is a stieltjes transform then f is reciprocally convex an application in probability is also presented

stochastic target problems dynamic programming and viscosity solutions in this paper we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward sdes and forward backward sdes the controlled process x nu y nu takes values in mathbb r d times mathbb r and a given initial data for x nu 0 then the control problem is to find the minimal initial data for y nu so that it reaches a stochastic target at a specified terminal time t the main application is from financial mathematics in which the process x nu is related to stock price y nu is the wealth process and nu is the portfolio we introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation the boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense this provides a unique characterization of the value function which is the minimal initial data for y nu

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

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

connection factors in the schrodinger equation with a polynomial potential given a second order differential equation with two singular points namely the origin and infinity the connection factors allow to split a power series solution into formal solutions with known asymptotic behavior a procedure is suggested to obtain those factors as quotients of wronskians of the mentioned solutions in the case of a schrodinger equation with a polynomial potential application of the procedure to particular cases whose connection factors are already known allows us to obtain new relations for quotients and products of gamma functions

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

some properties of functions related to the gamma psi and tetragamma functions in this paper some properties associated with the complete monotonicity and the logarithmically complete monotonicity of functions related to the gamma psi and tetragamma functions are obtained relevant connections of the results presented here with those derived in earlier works are also pointed out

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

new integrable quantum chains combining different kinds of spins a general procedure to generate new integrable hamiltonians combining to any kind of spins distributed arbitrarily on the line is given as a concrete application anisotropic chains formed by spin 1 2 and spin 1 operators at alternating sites are presented and solved exactly by the bethe ansatz ba the authors compute the ground state and excitation energies and momentum the higher order ba equations are derived depending on the choice these new hamiltonians exhibit or not conformal invariance in their low energy spectrum