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

convexity of quasi entropy type functions lieb s and ando s convexity theorems revisited given a positive function f on 0 and a non zero real parameter we consider a function if a b x tr x f larb 1 rb x in three matrices a b 0 and x this generalizes the notion of monotone metrics on positive definite matrices and in the literature 1 has been typical we investigate how operator monotony of f is sufficient and or necessary for joint convexity concavity of if a b x similar discussions are given for quasi entropies and quantum skew informations

inequalities and bounds for generalized complete elliptic integrals computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained also bounds for some combinations and products for integrals under discussion are established it has been proven that both families of integrals are logarithmically convex as functions of the first parameter this property has been employed to obtain several inequalities involving integrals in question

monotonicity properties and inequalities for the generalized elliptic integral of the first kind abstract the authors obtain some monotonicity and concavity properties and asymptotically sharp bounds for the generalized elliptic integral k a r of the first kind and its special case k r the complete elliptic integral of the first kind by studying the monotonicity and concavity properties of certain combinations defined in terms of the gaussian hypergeometric function f a 1 a 1 x and elementary functions thus improving some well known results

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

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

on certain inequalities for means ii a sequential method is applied to obtain inequalities between a mean introduced by h j seiffert and other means including the logarithmic mean the identric mean the arithmetic geometric mean of gauss and certain other means

dimension free quasiconformal distortion in n space most distortion theorems for k quasiconformal mappings in rn n 2 depend on both n and k in an essential way with bounds that become infinite as n tends to oo the present authors obtain dimension free versions of four well known distortion theorems for quasiconformal mappings namely bounds for the linear dilatation the schwarz lemma the e distortion theorem and the r1 quasisymmetry property of these mappings they show that the upper estimates they have obtained in each of these four main results remain bounded as n tends to oo with k fixed the proofs are based on a dimensioncancellation property of the function t h r 1 r t k t 0 k 0 where r t is the capacity of a teichmiiller extremal ring in rn the authors also prove a dimension free distortion theorem for the absolute cross ratio under k quasiconformal mappings of n from which several other distortion theorems follow as special cases

complete monotonicity and zeros of sums of squared baskakov functions we prove complete monotonicity of sums of squares of generalized baskakov basis functions by deriving the corresponding results for hypergeometric functions moreover in the central baskakov case we study the distribution of the complex zeros for large values of a parameter we finally discuss the extension of some results for sums of higher powers

constructing weyl group multiple dirichlet series let be a reduced root system of rank r a weyl group multiple dirichlet series for is a dirichlet series in r complex variables s1 sr initially converging for re si sufficiently large that has meromorphic continuation to c and satisfies functional equations under the transformations of c corresponding to the weyl group of a heuristic definition of such series was given in bbc06 and they have been investigated in certain special cases in bbc06 bbf06 bbfh07 bbf08 bbfa cg07 cfg08 cg chi05 in this paper we generalize results in cg to construct weyl group multiple dirichlet series by a uniform method and show in all cases that they have the expected properties

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

sharp distortion theorems for quasiconformal mappings continuing their earlier work on distortion theory the authors prove some dimension free distortion theorems for k quasiconformal mappings in r for example one of the present results is the following sharp variant of the schwarz lemma if f is a k quasiconformal self mapping of the unit ball bn n 2 with f 0 0 then 41 k2ixik i if x i 411 k2ixil k for all x in b

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

arithmetic geometric mean and related inequalities for operators abstract in recent years certain arithmetic geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on we first point out that they are direct consequences of integral expressions of relevant operators furthermore we obtain related new inequalities theorems 4 5 and 6 based on our current approach

a source of inequalities for circular functions in this work a basic theorem is established and it is found to be a source of inequalities for circular functions such as cusa s huygens wilker s the sandor bencze carlson s the shafer fink inequality and the one in the problem of oppenheim furthermore these inequalities described above will be extended by this basic theorem

families of completely positive maps associated with monotone metrics abstract an operator convex function on 0 which satisfies the symmetry condition k x 1 x k x can be used to define a type of non commutative multiplication by a positive definite matrix or its inverse using the primitive concepts of left and right multiplication and the functional calculus the operators for the inverse can be used to define quadratic forms associated with riemannian metrics which contract under the action of completely positive trace preserving maps we study the question of when these operators define maps which are also completely positive cp although a d 1 2 a d 1 2 is the only case for which both the map and its inverse are cp there are several well known one parameter families for which either the map or its inverse is cp we present a complete analysis of the behavior of these families as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points our primary tool is an order relation based on the concept of positive definite functions although some results can be obtained from known properties we also prove new results based on the positivity of the fourier transforms of certain functions concrete computations of certain fourier transforms not only yield new examples of positive definite functions but also examples in the much stronger class of infinitely divisible functions

turan type inequalities for generalized complete elliptic integrals in this paper our aim is to establish some turan type inequalities for gaussian hypergeometric functions and for generalized complete elliptic integrals these results complete the earlier result of p turan proved for legendre polynomials moreover we show that there is a close connection between a turan type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind at the end of this paper we prove a recent conjecture of t sugawa and m vuorinen related to estimates of the hyperbolic distance of the twice punctured plane

asymptotical bounds for complete elliptic integrals of the second kind abstract in this paper we establish several asymptotical bounds for the complete elliptic integrals of the second kind e r and improve the well known conjecture e r 1 1 r 2 3 4 2 2 3 2 for all r 0 1 proposed by m vuorinen

bounds for complete elliptic integrals of the second kind with applications in this paper we present some sharp bounds for complete elliptic integrals of the second kind as applications we establish several inequalities for the perimeter of an ellipse

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

submodular spectral functions of principal submatrices of a hermitian matrix extensions and applications we extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions we show that if f is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix a then the function i mapsto rm tr f a i is supermodular meaning that rm tr f a i rm tr f a j leq rm tr f a i cup j rm tr f a i cap j where a i denotes the i times i principal submatrix of a we discuss extensions to self adjoint operators on infinite dimensional hilbert space and to m matrices we discuss an application to cur approximation of nonnegative hermitian matrices

six new redheffer type inequalities for circular and hyperbolic functions in this paper six new redheffer type inequalities involving circular functions and hyperbolic functions 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

inequalities for sums of powers be valid for all n e n and x x 0 it is necessary ana s cient thut min a b min c d and max a 6 d max c d 5 in i brenner showed independently that 5 is a sufficient condition for 4 he also obtained a number of more general results there exist other generalizations of theorem a see for instance losonczi 3 pales s but now we show one way that can also be applied to the inequality 1 assume that a b and c d and then rearrange 4 to obtain l q xy h xf p h x xt xi i xf x i cd c it seems to be natural to consider the following more general inequality

concavity of the complete elliptic integrals of the second kind with respect to holder means abstract in this paper we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to holder means

inequalities for hyperbolic functions abstract several inequalities involving hyperbolic functions are derived some of them are obtained with the aid of stolarsky and gini means

generalized hersch pfluger distortion function and complete elliptic integrals abstract in this paper we study the relationship between the generalized hersch pfluger distortion function k a r and complete elliptic integrals of the second kind e r and establish several inequalities for k a r and e r

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

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

on a rankin selberg integral of the l function for widetilde mathrm sl _2 times mathrm gl _2 sl 2 gl2 we present a rankin selberg integral on the exceptional group g 2 which represents the l function for generic cuspidal representations of widetilde mathrm sl 2 times mathrm gl 2 as an application we show that certain fourier jacobi type periods on g 2 are non vanishing

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

functional equations involving means and gauss compositions of means abstract in this paper we give a review of results on functional equations involving means and their gauss compositions we also formulate some open problems

optimal evaluations of some seiffert type means by power means let us consider the second trigonometric mean t defined by seiffert and the hyperbolic mean m defined by neuman and sandor there are some known inequalities between these means and some power means a p we prove that the evaluationsa l n 2 l n l n 3 2 2

inequalities for means abstract a monotone form of l hospital s rule is obtained and applied to derive inequalities between the arithmetic geometric mean of gauss the logarithmic mean and stolarsky s identric mean some related inequalities are given for complete elliptic integrals

majorization and some operator monotone functions abstract let h t be a non decreasing function on i and k t an increasing function on j then h is said to be majorized by k if k a k b implies h a h b f t is operator monotone by definition if f t is majorized by t by making use of this majorization we will show that t a t b t r a r t 1 r b 1 r is operator monotone on 0 for 0 a b and for 0 r 1 the special case of a b 1 is the theorem due to petz hasegawa

monotonicity criterion for the quotient of power series with applications abstract in this paper we present the necessary and sufficient condition for the monotonicity of the quotient of power series as applications some gaps and misquotations in certain published articles are pointed out and corrected and some known results involving the landen inequalities for zero balanced hypergeometric functions are improved

the arithmetic geometric mean and fast computation of elementary functions we produce a self contained account of the relationship between the gaussian arithmetic geometric mean iteration and the fast computation of elementary functions a particularly pleasant algorithm for x is one of the by products

precise bounds for the weighted holder mean of the complete p elliptic integrals abstract in the article we present the best possible bounds for the weighted holder mean of the complete p elliptic integrals of the first and second kinds which are the generalizations of the previously results for the complete elliptic integrals

on approximating the arithmetic geometric mean and complete elliptic integral of the first kind abstract in the article we prove that the double inequalities 1 6 p 7 r p 5 p 6 r tanh 1 r 2 r k r 1 6 q 7 r q 5 q 6 r tanh 1 r 2 r q a 1 r 5 q 6 g 1 r a 1 r 6 q 7 g 1 r l 1 r a g m 1 r p a 1 r 5 p 6 g 1 r a 1 r 6 p 7 g 1 r l 1 r hold for all r 0 1 if and only if p 2 1 570796 and q 89 69 1 289855 where k r 0 2 1 r 2 sin 2 t 1 2 d t is the complete elliptic integral of the first kind tanh 1 r log 1 r 1 r 2 is the inverse hyperbolic tangent function r 1 r 2 and a 1 r 1 r 2 g 1 r r l 1 r r 1 log r and a g m 1 r are the arithmetic geometric logarithmic and gaussian arithmetic geometric means of 1 and r respectively

computer aided solution of the invariance equation for two variable stolarsky means abstract we solve the so called invariance equation in the class of two variable stolarsky means s p q p q r i e we find necessary and sufficient conditions on the six parameters a b c d p q such that the identity s p q s a b x y s c d x y s p q x y x y r be valid we recall that for pq p q 0 and x y the stolarsky mean s p q is defined by s p q x y q x p y p p x q y q 1 p q in the proof first we approximate the stolarsky mean and we use the computer algebra system maple v release 9 to compute the taylor expansion of the approximation up to 12th order which enables us to describe all the cases of the equality

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

schur convexity and schur geometrically concavity of gini means the schur convexity and the schur geometric convexity with variables x y r 2 for fixed s t of gini means g r s x y are discussed some new inequalities are obtained

inequalities for compound mean iterations with logarithmic asymptotes abstract we consider the compound means arising as limits from the arithmetic geometric mean iteration and related iterations each of these iterations possesses a logarithmic asymptote we show that these limit means satisfy very precise inequalities these can be deduced in a quite uniform fashion from a comparison lemma for compound means

a metaplectic casselman shalika formula for gl r we provide formulas for various bases of spherical whittaker functions on the n fold metaplectic cover of rm gl r over a p adic field and show that there is a basis of symmetric functions in the complex parameter in addition we relate a specific spherical whittaker function to the p power part of the weyl group multiple dirichlet series for the root system of type a r 1 constructed from n th order gauss sums we also show that the zonal spherical functions can be computed explicitly in terms of hall littlewood polynomials as in macdonald s formula for rm gl r

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

sharp power mean bounds for the gaussian hypergeometric function sharp inequalities are established between the gaussian hypergeometric function and the power mean these results extend known inequalities involving the complete elliptic integral and the hypergeometric mean

harmonic maps of s2 into a complex grassmann manifold abstract let g k n be the grassmann manifold of all ck in cn the complex spaces of dimensions k and n respectively or what is the same the manifold of all projective spaces pk 1 in pn 1 so that g 1 n is the complex projective space pn 1 itself we study harmonic maps of the two dimensional sphere s2 into g k n the case k 1 has been the subject of investigation by several authors see for example din a m zakrzewski w j 1980 nucl phys b 174 397 406 eells j wood j c 1983 adv math 49 217 263 and wolfson j g trans am math soc in press the harmonic maps s2 g 2 4 have been studied by ramanathan ramanathan j 1984 j differ geom 19 207 219 we shall describe all harmonic maps s2 g 2 n the method is based on several geometrical constructions which lead from a given harmonic map to new harmonic maps in which the image projective spaces are related by fundamental collineations the key result is the degeneracy of some fundamental collineations which is a global consequence following from the fact that the domain manifold is s2 the method extends to g k n

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

explicit construction of harmonic two spheres into the complex grassmannian we present an explicit description of all harmonic maps of finite uniton number from a riemann surface into a complex grassmannian namely starting from a constant map q and a collection of meromorphic functions and their derivatives we show how to algebraically construct all harmonic maps from the two sphere into a given grassmannian g p mathbb c n in this setting the uniton number depends on q and p and we obtain a sharp estimate for it

sharp approximations for the complete elliptic integrals of the second kind by one parameter means abstract let e r be the complete elliptic integrals of the second kind and j p a b be the one parameter mean of two distinct positive numbers a and b defined by j p a b p p 1 a p 1 b p 1 a p b p for p 1 0 in this study using the recurrence method with the sign rule of a special power series we prove that the double inequality 7 22 j 2 2 1 r 2 e r j 2 2 1 r holds for r 1 r 2 0 1 where the upper bound is sharp while the lower bound is related to the archimedes approximation 22 7 for using the same method we also easily show that both of the functions r 2 2 e r j 7 4 1 r r 12 and r 2 1 r 10 1 1 2 e r 1 j 7 4 1 r are absolutely monotonic on 0 1 which greatly improves the known results moreover it should be emphasized that our method may be used to explore special functions related to hypergeometric functions

completely positive mappings and mean matrices abstract some functions f r r induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices moreover such a function f can define a linear mapping j d f 1 m n m n on matrices which is basic in the constructions of monotone metrics the present subject is to check the complete positivity of j d f 1 in the case of a few concrete functions f this problem has been motivated by applications in quantum information

some results in the theory of genuine representations of the metaplectic double cover of gsp2n f over p adic fields abstract let f be a p adic field and let g n and g 0 n be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over f we show here that if n is odd then all the genuine irreducible representations of g n are induced from a normal subgroup of finite index closely related to g 0 n thus we reduce in this case the theory of genuine admissible representations of g n to the better understood corresponding theory of g 0 n for odd n we also prove the uniqueness of certain whittaker functionals along with rodier type of heredity our results apply also to all parabolic subgroups of g n if n is odd and to some of the parabolic subgroups of g n if n is even we prove some irreducibility criteria for parabolic induction on g n for both even and odd n as a corollary we show among other results that while for odd n all genuine principal series representations of g n induced from unitary representations are irreducible there exist reducibility points on the unitary axis if n is even we also list all the reducible genuine principal series representations of g 2 provided that the f is not 2 adic

functional inequalities for modified struve functions in this paper by using a general result on the monotonicity of quotients of power series our aim is to prove some monotonicity and convexity results for the modified struve functions moreover as consequences of the above mentioned results we present some functional inequalities as well as lower and upper bounds for modified struve functions our main results complement and improve the results of joshi and nalwaya