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

refinements and extensions of an inequality iii abstract in this article all of the cases of the inequality formula being or not being valid are discussed for all real numbersx y r andx 0 wherea b 0 anda b some recent developments concerning mean values are simply introduced

schauder estimates for degenerate monge ampere equations and smoothness of the eigenfunctions we obtain c 2 beta estimates up to the boundary for solutions to degenerate monge ampere equations of the type begin aligned det d 2 u f text in omega quad f sim text dist alpha cdot partial omega text near partial omega alpha 0 end aligned as a consequence we obtain global c infty estimates up to the boundary for the eigenfunctions of the monge ampere operator det d 2 u 1 n on smooth bounded uniformly convex domains in mathbb r n

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

radially symmetric boundary value problems for real and complex elliptic monge ampere equations abstract radially symmetric dirichlet and neumann problems for real and complex monge ampere equations are considered existence of radially symmetric solutions is proved by transforming the differential equations into integral ones solvable by means of fixed point arguments then taking advantage of integral formulae regularity and convexity of the radial solutions are checked fairly weak assumptions are required in that process in the real case a priori radial symmetry is also discussed

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

extended means as weighted means extended means can be viewed as special cases of the well known weighted means with this point of view many properties of the extended means like monotonicity and convexity can be derived very easily from the corresponding established results of the weighted means

on mod p properties of siegel modular forms we prove two results on mod p properties of siegel modular forms first we use theta series in order to construct of a siegel modular form of weight p 1 which is congruent to 1 mod p second we define a theta operator vartheta on q expansions and show that the algebra of siegel modular forms mod p is stable under vartheta by exploiting the relation between vartheta and generalized rankin cohen brackets

a simple proof of monotonicity for extended mean values abstract in the note a simple and elementary proof of monotonicity is given for the so called extended mean values

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

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

integral representations and complete monotonicity related to the remainder of burnside s formula for the gamma function abstract in the paper the authors establish integral representations of some functions related to the remainder of burnside s formula for the gamma function and find the logarithmically complete monotonicity of these and related functions these results extend and generalize some known conclusions

on the local and global comparison of generalized bajraktarevic means abstract given two continuous functions f g i r such that g is positive and f g is strictly monotone a measurable space t a a measurable family of d variable means m i d t i and a probability measure on the measurable sets a the d variable mean m f g m i d i is defined by m f g m x f g 1 t f m x 1 x d t d t t g m x 1 x d t d t x x 1 x d i d the aim of this paper is to study the local and global comparison problem of these means i e to find conditions for the generating functions f g and h k for the families of means m and n and for the measures such that the comparison inequality m f g m x m h k n x x i d be satisfied

some complete monotonicity properties for the p q gamma function for the c p q function we derive several properties and characteristics related to convexity log convexity and complete monotonicity similar properties and characteristics of the corresponding p q analogue j p q x of the digamma or the j function have also been established by applying the main results in this paper when p and q 1 we obtain all of the results given in several earlier works by for example krasniqi shabani and other authors some potential areas of applications of the results presented in this paper are also indicated

multi variable extended mean values on considere des fonctions dont les valeurs sont des valeurs moyennes d une suite x x 0 x n de nombre reels pour une bonne paire f g de fonctions on definit une valeur moyenne e f g x de chaque suite x a valeurs dans cette intervalle

a new lower bound in the second kershaw s double inequality in the paper a new and elegant lower bound in the second kershaw s double inequality is established some alternative simple and polished proofs are given several deduced functions involving the gamma and psi functions are proved to be decreasingly monotonic and logarithmically completely monotonic and some remarks and comparisons are stated

boundary behavior of large solutions to the monge ampere equations with weights abstract this paper is concerned with the monge ampere equation det d 2 u x b x f u x x where is a strictly convex bounded smooth domain in r n with n 2 and b c which is positive in but may be vanishing on the boundary we find a new structure condition on f which plays a crucial role in the boundary behavior of strictly convex large solutions our results are obtained in a more general setting than those in cirstea and trombetti 2008 12 where f is regularly varying at infinity with index p n

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

on singular boundary value problems for the monge ampere operator abstract we consider different types of singular boundary value problems for the monge ampere operator the approach is based on existing regularity theory and a subsolution supersolution method nonexistence and uniqueness results are also given

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

existence and estimates of solutions to a singular dirichlet problem for the monge ampere equation given a strictly convex smooth and bounded domain in rn we establish the existence of a negative convex solution in c c with zero boundary value to the singular monge ampere equation det d2u p x g u an associated dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem estimates of solutions will also be given and regularity of solutions will be deduced from the estimates

means that minimize relative error and an associated integral equation abstract determinations are made of the means that minimize various relative errors in the sense that the harmonic mean of a and b minimizes the traditional relative error on a b the general problem for averaged relative error leads to a nonlinear integral equation for which we prove existence and uniqueness results as well as constructive solution procedures the inverse problem of finding the measure of relative error corresponding to a given mean is also analyzed in detail these studies shed light on both new and previously known inequalities for specific means

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

integral representations of bivariate complex geometric mean and their applications abstract in the paper the authors survey integral representations including the levy khintchine representations and applications of some bivariate means including the logarithmic mean the identric mean stolarsky s mean the harmonic mean the weighted geometric means and their reciprocals and the toader qi mean and the multivariate weighted geometric means and their reciprocals derive integral representations of bivariate complex geometric mean and its reciprocal and apply these newly derived integral representations to establish integral representations of heronian mean of power 2 and its reciprocal

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

note on mod p siegel modular forms ii the structure of the ring of mod p siegel modular forms of degree two is determined in the cases where the prime p is 2 or 3

a new upper bound in the second kershaw s double inequality and its generalizations in the paper a new upper bound in the second kershaw s double inequality involving ratio of gamma functions is established and as generalizations of the second kershaw s double inequality the divided differences of the psi and polygamma functions are bounded

covariants of binary sextics and vector valued siegel modular forms of genus two we extend igusa s description of the relation between invariants of binary sextics and siegel modular forms of degree 2 to a relation between covariants and vector valued siegel modular forms of degree 2 we show how this relation can be used to effectively calculate the fourier expansions of siegel modular forms of degree 2

note on monotonicity of generalized weighted mean values in the note using the cauchy schwarz buniakowski inequality the authors prove that the generalized weighted mean values mp f r s x y are increasing with respect to the two parameters r and s for any given non negative weight p and positive function f

blow up rates of large solutions for elliptic equations in this paper we mainly study the boundary behavior of solutions to boundary blow up elliptic problems for more general nonlinearities f which may be rapidly varying at infinity u b x f u x u where is a bounded domain with smooth boundary in rn and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary further when f s sp f1 s for s sufficiently large where p 1 and f1 is normalized regularly varying at infinity with index p1 0 p we show the influence of the geometry of on the boundary behavior for solutions to the problem we also give the existence and uniqueness of solutions

some properties of central delannoy numbers abstract in the paper by investigating the generating function of central delannoy numbers the authors establish several explicit expressions including determinantal expressions for central delannoy numbers present three identities involving the cauchy products of central delannoy numbers discover an integral representation for central delannoy numbers find absolute monotonicity convexity and logarithmic convexity for the sequence of central delannoy numbers and construct several product and determinantal inequalities for central delannoy numbers

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

optimal boundary regularity for a singular monge ampere equation abstract in this paper we study the optimal global regularity for a singular monge ampere type equation which arises from a few geometric problems we find that the global regularity does not depend on the smoothness of domain but it does depend on the convexity of the domain we introduce a type to describe the convexity as a result we show that the more convex is the domain the better is the regularity of the solution in particular the regularity is the best near angular points

the reciprocal of the geometric mean of many positive numbers is a stieltjes transform in the paper by the cauchy integral formula in the theory of complex functions an integral representation for the reciprocal of the geometric mean of many positive numbers is established as a result the reciprocal of the geometric mean of many positive numbers is verified to be a stieltjes transform and consequently a logarithmically completely monotonic 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

large solutions to the monge ampere equations with nonlinear gradient terms existence and boundary behavior abstract in this paper we obtain conditions about the existence and boundary behavior of strictly convex solutions to the monge ampere equations with boundary blow up det d 2 u x b x f u x u q x u and det d 2 u x b x f u x 1 u q x u where is a strictly convex bounded smooth domain in r n with n 2 q 0 n or q 0 n b c which is positive in but may vanish or blow up on the boundary f c 0 f 0 0 and f is strictly increasing on 0 or f c r f s 0 s r and f is strictly increasing on r

the ring of modular forms of degree two in characteristic three we determine the structure of the ring of siegel modular forms of degree 2 in characteristic 3

on jessen s inequality for convex functions ii on etablit quelques resultats lies a l integralite de b jessen pour les fonctions convexes

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

a class of logarithmically completely monotonic functions and application to the best bounds in the second gautschi kershaw s inequality in this article the logarithmically complete monotonicity of the function c x b c x a 1 a b exp j x c are discussed where a b c are real numbers and c is the classical euler s gamma function from this the best upper and lower bounds for walls ratio c x 1 c x s are established which refine the second gautschi kershaw s inequality

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

two optimal double inequalities between power mean and logarithmic mean for email protected r the power mean m p a b of order p the logarithmic mean l a b and the arithmetic mean a a b of two positive real values a and b are defined by m p a b a p b p2 1 p p 0 ab p 0 l a b b alogb loga a b a a b and a a b a b2 respectively in this article we answer the questions what are the greatest values p and r and the least values q and s such that the inequalities m p a b 13a a b 23l a b m q a b and m r a b 23a a b 13l a b m s a b hold for all a b 0

comparison of means generated by two functions and a measure abstract given two continuous functions f g i r such that g is positive and f g is strictly monotone and a probability measure on the borel subsets of 0 1 the two variable mean m f g i 2 i is defined by m f g x y f g 1 0 1 f t x 1 t y d t 0 1 g t x 1 t y d t x y i the aim of this paper is to study the comparison problem of these means i e to find conditions for the generating functions f g and h k and for the measures such that the comparison inequality m f g x y m h k x y x y i holds

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

boundary behavior for large solutions to elliptic equations with singular weights abstract in this paper we analyze the boundary behavior of large positive solutions to some semilinear elliptic equations which include a singular weight the most important point is that the growth of the solutions can be determined in terms of the solution to a one dimensional first order equation we also consider the questions of existence and uniqueness of positive solutions

optimal global and boundary asymptotic behavior of large solutions to the monge ampere equation abstract this paper is mainly concerned with optimal global and boundary asymptotic behavior of strict convex large solutions to the monge ampere equation det d 2 u b x f u x where is a strict convex and bounded smooth domain in r n with n 2 f c 1 0 or f c 1 r which is increasing in 0 or r and satisfies the keller osserman type condition b c is positive in but may vanish or blow up on the boundary properly we find new structure conditions on f which play a crucial role in both global and boundary behavior of such solutions moreover we reveal asymptotic behavior of such solutions when the parameters on b tend to the corresponding critical values in addition when f does not satisfy the keller osserman type condition and is a ball we supply a necessary and sufficient condition on b for the existence of an infinitude of strict convex radially symmetric positive solutions to such problem

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

inequalities for differences of powers on etudie l inegalite 1 x a 1 y a 1 a 1 1 x a k y a k a k k avec 0 1 k on deduit des conditions necessaires de validite pour tout x y 0 avec x y on demontre que ces conditions necessaires sont aussi suffisantes

the existence and asymptotic behavior of boundary blow up solutions to the k hessian equation abstract in this paper we consider the existence and asymptotic behavior of k convex solution to the boundary blow up k hessian problem s k d 2 u h x f u for x u x as dist x 0 where k 1 2 n s k d 2 u is the k hessian operator is a smooth bounded strictly convex domain in r n n 2 h c is positive in but is not necessarily bounded on and f is a smooth positive function that satisfies the so called keller osserman condition further results are obtained for the special case that is a ball our approach to show the existence and asymptotic behavior exploits the method of sub and super solutions and karamata regular variation theory

monotonicity theorems and inequalities for the complete elliptic integrals we prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind k and e these results lead to sharp symmetrical bounds for k and e which improve recently discovered inequalities

existence and boundary asymptotic behavior of large solutions of hessian equations abstract in this paper we establish the existence of large solutions of hessian equations and obtain a new boundary asymptotic behavior of solutions

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

the hasegawa petz mean properties and inequalities abstract we study a family of means introduced by h hasegawa and d petz 1996 13 properties with respect to the parameter such as monotonicity and logarithmic concavity further monotonicity and concavity in the mean variables are shown besides the comparison between the hasegawa petz mean and the geometric mean is completely solved the connection to earlier results on operator monotonicity and some applications are also discussed

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 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