hypergeometric transforms of functions with derivative in a half plane let a be the class of normalized analytic functions in the unit disk f a b c z and a c z denote respectively the gaussian and confluent hypergeometric functions let r lcub a r rcub such that re ei z 0 z rcub for a we define the hypergeometric transforms va b c and ua c by the convolution va b c zf a b c z z and ua c z a c z z respectively the main aim of this paper is to find conditions on 1 2 and the parameters a b c such that each of the operators va b c and ua c maps r 1 into r 2 we also find conditions such that the fu cab f a b c z 1 or ca a c z 1 is in r

geometrical implications of certain infinite dimensional decompositions we investigate the connections between the global structure of a banach space i e the existence of certain finite and infinite dimensional de compositions and the geometrical properties of the closed convex bounded sub sets of such a space i e the existence of extremal and other topologically distin guished points the global structures of various supposedly pathological examples of banach spaces constructed by r c james turn out to be more universal than expected for instance james tree type resp james matrix type decompositions characterize banach spaces with the point of continuity property resp the radon nikodym property moreover the convex point of continuity property is stable under the formation of james infinitely branching tree type sums of infinite dimensional factors we also give several counterex amples to various questions relating some topological and geometrical concepts in banach spaces

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

polynomial series expansions for confluent and gaussian hypergeometric functions based on the hadamard product of power series polynomial series expansions for confluent hypergeometric functions m a c and for gaussian hypergeometric functions f a b c are introduced and studied it turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions m a c and f a b c in particular if the parameters are also viewed as variables

distribution of growth probabilities in fluid flow and diffusion limited aggregation the authors examine the role of extrinsic noise in diffusion limited aggregation dla and a deterministic continuum theory derived from the work of paterson 1984 on fluid flow in a hele shaw cell they show the distribution of growth probabilities for the deterministic model to be essentially the same as dla and predict that the asymptotic value of the fractal dimension for off lattice dla may be closer to d approximately 1 65 or 0 03 as opposed to d 1 71 the results confirm that extrinsic noise is not essential to create the characteristic properties of dla in models that are variations of dla

mean field diffusion limited aggregation a density model for viscous fingering phenomena we explore a universal density formalism to describe nonequilibrium growth processes specifically the immiscible viscous fingering in hele shaw cells usually referred to as the saffman taylor problem for that we develop an alternative approach to the viscous fingering phenomena whose basic concepts have been recently published in a rapid communication phys rev e 63 045305 r 2001 this approach uses the diffusion limited aggregation dla paradigm as a core we introduce a mean field dla generalization in stochastic and deterministic formulations the stochastic model a quasicontinuum dla simulates monte carlo patterns which demonstrate a striking resemblance to natural hele shaw fingers and for steady state growth regimes follow precisely the saffman taylor analytical solutions in channel and sector configurations the relevant deterministic theory a complete set of differential equations for a time development of density fields is derived from that stochastic model as a principal conclusion we prove an asymptotic equivalency of both the stochastic and deterministic mean field dla formulations to the classic saffman taylor hydrodynamics in terms of an interface evolution

symmetrically separated sequences in the unit sphere of a banach space abstract we prove the symmetric version of kottman s theorem that is to say we demonstrate that the unit sphere of an infinite dimensional banach space contains an infinite subset a with the property that x y 1 for distinct elements x y a thereby answering a question of j m f castillo in the case where x contains an infinite dimensional separable dual space or an unconditional basic sequence the set a may be chosen in a way that x y 1 e for some e 0 and distinct x y a under additional structural properties of x such as non trivial cotype we obtain quantitative estimates for the said e certain renorming results are also presented

on hypergeometric functions and function spaces the aim of this paper is to discuss the role of hypergeometric functions in function spaces and to prove some new results for these functions the first part of this paper proves results such as monotone convexity and concavity properties of sums of products of hypergeometric functions the second part of our results deals with the space a of all normalized analytic functions f f 0 0 f 0 1 in the unit disk and the subspace r f a a n r such that re ei f z 0 z for f a we consider integral transforms of the type v f 01 t f tz t dt where t is a real valued nonnegative weight function normalized so that 01 t 1 we obtain conditions on and the function such that v f takes each member of r into a starlike function of order 0 1 2 these results extend and improve the earlier known results in these directions we end the paper with an open problem

generalized elliptic integrals and the legendre m function abstract we study monotonicity and convexity properties of functions arising in the theory of elliptic integrals and in particular in the case of a schwarz christoffel conformal mapping from a half plane to a trapezoid we obtain sharp monotonicity and convexity results for combinations of these functions as well as functional inequalities and a linearization property

the analytic complete continuity property abstract we introduce the notion of the analytic complete continuity property of banach spaces we give different characterizations of this property we show that this property is different from known related properties such as the complete continuity property and the analytic radon nikod m property

a study of conditional spreading sequences abstract it is shown that every conditional spreading sequence can be decomposed into two well behaved parts one being unconditional and the other being convex block homogeneous i e equivalent to its convex block sequences this decomposition is then used to prove several results concerning the structure of spaces with conditional spreading bases as well as results in the theory of conditional spreading models among other things it is shown that the space c is universal for all spreading models i e it admits all spreading sequences both conditional and unconditional as spreading models moreover every conditional spreading sequence is generated as a spreading model by a sequence in a space that is quasi reflexive of order one

integral transforms of certain subclasses of analytic functions abstract let a be the class of normalized analytic functions in the unit disk and define the class p f a r re e i 1 f z z f z 0 z for a function f a we define the integral transform v f z 1 0 t f tz t dt where is a real valued nonnegative weight function with 10 t dt 1 for f p we will find conditions such that v f is starlike and also such that v f p we also discuss how to write the integral transform as a convolution with a 2f1 hypergeometric function

a classification of baire class 1 functions we study in this paper various ordinal ranks of bounded baire class 1 functions and we show their essential equivalence this leads to a natural classification of the class of bounded baire class 1 functions b 1 in a transfinite hierarchy b 1 1 of small baire classes for which for example an analysis similar to the hausdorff kuratowski analysis of 0 2 sets via transfinite differences of closed sets can be carried out the notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a baire class 1 function are introduced and used in this study

applications of the theory of semi embeddings to banach space theory let x and y be banach spaces and t x y an injective bounded linear operator t is called a semi embedding if t maps the closed unit ball of x to a closed subset of y this concept was introduced by lotz peck and porta proc edinburgh math soc 22 1979 233 240 it is proved that if x semi embeds in y and x is separable then x has the radon nikodym property provided y does it is shown that if l1 semi embeds in y then y fails the schur property and contains a subspace isomorphic to l1 as a consequence of the proof it is shown that if x is a subspace of l1 either l1 embeds in x or l1 embeds in l1x the simpler result that l1 does not semi embed in c0 is treated separately this result is used to deduce the classic result of menchoff that there exists a singular probability measure on the circle with fourier coefficients vanishing at infinity some generalizations of the notion of semi embedding are given and several complements and open questions are discussed

borel properties of linear operators abstract given an injective bounded linear operator t x y between banach spaces we study the borel measurability of the inverse map t 1 tx x a remarkable result of saint raymond ann inst fourier grenoble 26 1976 211 256 states that if x is separable then the borel class of t 1 is if and only if x is the th iterated sequential weak closure of t y for some countable ordinal we show that saint raymond s result holds with minor changes for arbitrary banach spaces if we assume that t has certain property named co discreteness after hansell proc london math soc 28 1974 683 699 as an application we show that the borel class of the inverse of a co discrete operator t can be estimated by the image of the unit ball or the restrictions of t to separable subspaces of x our results apply naturally when x is a wcd banach space since in this case any injective bounded linear operator defined on x is automatically co discrete

functional inequalities for the ratio of complete p elliptic integrals abstract in this paper we show monotonicity and convexity properties of a function involving the ratio of complete p elliptic integrals which reduces to the modulus of the grotzsch ring as applications we obtain several sharp functional inequalities

a characterization of banach spaces containing it is proved that a banach space contains a subspace isomorphic to 11 if and only if it has a bounded sequence with no weak cauchy subsequence the proof yields that a sequence of subsets of a given set has a subsequence that is either convergent or boolean independent a bounded sequence of elements fn in a banach space b is said to be equivalent to the usual 11 basis provided there is a a 0 so that for all n and choices of scalars c cn

solution to a conjecture on the legendre m function with an application to the generalized modulus abstract in this note we give a solution to a conjecture on the legendre m function proposed by heikkala linden vamanamurthy and vuorinen in the paper heikkala et al 2008 5 as an application of this result we obtain a functional inequality of the generalized modulus

weak polish banach spaces abstract a banach space x which is a subspace of the dual of a banach space y is said to be weak polish proviced x is separable and the closed unit ball of x is polish in the weak topology x is said to be polish provided it is weak polish when regarded as a subspace of x a variety of structural results are obtained for banach spaces with the point of continuity property pcp using the unifying concept of weak polish banach spaces it is proved that every weak polish banach space contains an infinite dimensional weak closed subspace this yields the result of edgar wheeler that every polish banach space has a reflexive subspace it also yields the result of ghoussoub maurey that every banach space with the pcp has a subspace isomorphic to a separable dual in virtue of the following known result linking the pcp and weak polish banach spaces a separable banach space has the pcp if edgar wheeler and only if ghoussoub maurey it is isometric to a weak polish subspace of the dual of some separable banach space for completeness a proof of this result is also given it is proved that if x and y are banach spaces with x y then x is weak polish if and only if x has a weak continuous boundedly complete skipped blocking decomposition combined with the above mentioned linkage this characterization of weak polish banach spaces gives the known result that a separable banach space has the pcp if bourgain rosenthal and only if ghoussoub maurey it has the boundedly complete skipped blocking property the proof of the characterization is a variation of the original arguments for this result the characterization also yields that a banach space is polish if and only if it has the reflexive skipped blocking property it is moreover demonstrated that a banach space has the pcp provided it has the pc skipped blocking property in particular this yields an exposition of the result of bourgain that a space has the pcp provided each of its subspaces with an fdd has the pcp it is proved that the pcp is a three space property thus completing the solution to a problem posed by edgar wheeler that is if y and x are banach spaces with y x then x has the pcp provided both y and x y have the pcp it is also shown that a banach space x has a separable dual provided it has the dual separable skipped blocking property the proof yields the dividend that x is separable provided x j is separable for every basic sequence x j in x

spaces of diagonal operators properties of space of diagonal operators on semi normalized schauder bases are studied a necessary and sufficient condition is given for the containment of c 0 in the space of compact diagonal operators on a semi normalized schauder basis using this condition it is shown that there exist semi normalized schauder bases of reflexive banach spaces such that c 0 does not embed in the space of compact diagonal operators on these schauder bases

cones with bounded and unbounded bases and reflexivity abstract in this paper we prove two characterizations of reflexivity for a banach space x the first one is based on the existence in x of a closed convex cone with a nonempty interior such that all the bases generated by a strictly positive functional are bounded while the second one is stated in terms of the nonexistence of a cone such that it has bounded and unbounded bases both generated by strictly positive functionals simultaneously we call such a cone mixed based cone we study the features of this class of cones in particular we show that every cone conically isomorphic to the nonnegative orthant l 1 of l 1 is a mixed based cone and that every mixed based cone contains a conically isomorphic copy of l 1 moreover we give a detailed description of the structure of a mixed based cone this approach allows us to prove some results concerning the embeddings of l 1 and c 0 in a banach space

the active perimeter in cluster growth models a rigorous bound a definition of an active perimeter is introduced for arbitrary cluster growth models a general bound is derived showing that this perimeter cannot grow faster than ra 1 where a df is the fractal dimension of the cluster for diffusion limited aggregates this is seen to be smaller than the growing interface as defined by meakin and witten 1983 a further consequence is that the eden model 1961 is fully compact in any dimension d i e it has a negligible number of internal holes further one finds that its perimeter scales as volume d 1 d

hausdorff measures and functions of bounded quadratic variation to each function f of bounded quadratic variation we associate a hausdorff measure f we show that the map f f is locally lipschitz and onto the positive cone of m 0 1 we use the measures f f v 2 to determine the structure of the subspaces of v 0 2 which either contain c 0 or the square stopping time space s 2

on a problem of h p rosenthal concerning operators on c 0 1 a problem of h p rosenthal asks whether every bounded linear operator which is an isomorphism on a closed linear infinite dimensional subspace x not containing any isomorph of c0 is actually an isomorphism on a subspace isomorphic to c 0 1 an affirmative answer to this problem is provided when t is a contraction whose restriction to x is an isometry

on functions of finite baire index it is proved that every function of finite baire index on a separable metric space k is a d function i e a difference of bounded semi continuous functions on k in fact it is a strong d function meaning it can be approximated arbitrarily closely in d norm by simple d functions it is shown that if the n th derived set of k is non empty for all finite n there exist d functions on k which are not strong d functions further structural results for the classes of finite index functions and strong d functions are also given

on hypergeometric functions and generalizations of legendre s relation we prove some convexity properties for a sum of hypergeometric functions and obtain a generalization of legendre s relation for complete elliptic integrals we apply these results to prove some inequalities for hypergeometric functions incomplete beta functions and legendre functions

an extension of a structure theorem of bourgain abstract bourgain has shown that if x is a banach space which does not have the radon nikodym property respectively the point of continuity property then x has a subspace y which also fails to have the respective property but which has a finite dimensional schauder decomposition we extend this result to spaces x failing the convex point of continuity property

decomposing borel sets and functions and the structure of baire class 1 functions all spaces considered are metric separable and are denoted usually by the letters x y or z w stands for the set of all natural numbers if a metric separable space is additionally complete we call it polish if it is a continuous image of ww or equivalently of a polish space it is called souslin the main subject of the present paper is the structure of baire class 1 functions recent developments in banach space theory in particular discoveries of rosenthal and bourgain fremlin and talagrand see r for a survey of these developments stimulated investigations into the structure of baire class 1 functions the interested reader may consult papers by haydon odel rosenthal hor kechris louveau kl rosenthal r2 and references quoted there these investigations have already provided new applications in banach space theory see for example ri in our study of baire class 1 functions we will be interested in two decomposition properties one of them defined by lusin the other one by jayne and rogers first however we want to consider a more general problem of determining how difficult it is to represent a borel set as a union of simpler borel sets or the graph of a borel function as a union of the graphs of simpler borel functions using effective descriptive set theory in particular louveau s theorem we show that if a c x is borel x polish then a e e or there is a continuous injection q ww a such that q 1 b is meager for any b c a which is e2 this gives a new proof of j stern s result that if a borel set a is the union of cov m sets in e then a is itself e cov m is the smallest cardinality of a family of meager sets covering ill we prove similar results for functions put for f x y and a family of functions g

gδ embeddings in hilbert space ii in this paper which is a continuation of 10 we exhibit some topological conditions on a banach space which ensure that it contains isometric copies of infinite dimensional conjugate spaces this result is used to identify a large class of banach spaces that are hereditarily separable duals a method of defining a jamestree sum of a countable number of banach spaces is given it is used to construct various counterexamples for instance there exists for each integer n a banach space that can be mapped into hilbert space via the composition of n but not n 1 g embeddings we also continue the investigation of the global structure of some geometrically defined banach spaces for example it is shown that a separable banach space x with the radon nikodym property r n p has a subspace y with a boundedly complete finite dimensional decomposition f d d such that xy has an f d d and the r n p

elliott s identity and hypergeometric functions elliott s identity involving the gaussian hypergeometric series contains as a special case the classical legendre identity for complete elliptic integrals the aim of this paper is to derive a differentiation formula for an expression involving the gaussian hypergeometric series which for appropriate values of the parameters implies elliott s identity and which also leads to concavity convexity properties of certain related functions we also show that elliott s identity is equivalent to a formula of ramanujan on the differentiation of quotients of hypergeometric functions applying these results we obtain a number of identities associated with the legendre functions of the first and the second kinds respectively

on the conditional expectation and convergence properties of random sets in this paper we study random sets with values in a separable banach space first we establish several useful properties of the set valued conditional expectation and then prove some convergence theorems for set valued amarts and uniform amarts using the weak kuratowski mosco and hausdorff modes of set convergence

gδ embeddings in hilbert space abstract it is shown that a separable banach space x has the point of weak to norm continuity property resp the radon nikodym property if and only if there exists a compact g embedding resp an h embedding from x into l2 this solves several questions of j bourgain and h p rosenthal j funct anal 52 1983 it is also shown that every non relatively compact sequence in a banach space with property pc has a difference subsequence which is a boundedly complete basic sequence this solves a question of pelczynski and extends some results of w b johnson and h p rosenthal studia math 43 1972 77 92 various related questions asked in the above bourgain rosenthal reference and by g a edgar and r f wheeler pac j math 115 1984 and n ghoussoub and h p rosenthal math ann 264 1983 321 332 are also settled

on baire 1 4 functions we give descriptions of the spaces d k i e the space of differences of bounded semicontinuous functions on k and especially of b1 4 k defined by haydon odell and rosenthal as well as for the norms which are defined on them for example it is proved that a bounded function on a metric space k belongs to b1 4 k if and only if the th oscillation osc f of f is bounded and in this case f 1 4 f osc f also we classify b1 4 k into a decreasing family s k 1 1 of banach spaces whose intersection is equal to d k and s1 k b1 4 k these spaces are characterized by spreading models of order equivalent to the summing basis of c0 and for every function f in s k it is valid that osc f is bounded finally using the notion of null coefficient of order sequence we characterize the baire 1 functions not belonging to s k introduction in recent years the study of the first baire class b1 k of bounded functions on a metric space k led to the definition of interesting subclasses h o r k l f1 the study of these subclasses revealed significant properties of their elements c m r r2 f1 f2 and provided applications such as the c0 dichotomy theorem of rosenthal r1 here we study some subclasses of d k and especially b1 4 k of b1 k by d k is denoted the class of all functions on k which are differences of bounded semicontinuous functions a classical result of baire yields that f d k if and only if there exists a sequence fn of continuous functions on k satisfying sup x k n fn x and f

univalence criteria and quasiconformal extensions let f be a locally univalent meromorphic function in the unit disk a recently epstein obtained a differential geometric proof for the fact that if f satisfies an inequality involving a suitable real valued function a then f is univalent in a and has a quasiconformal extension to the sphere we give a more classical proof for this result by means of an explicit quasiconformal extension and obtain generalizations of the result under suitable conditions even if a is allowed to be complex valued and a is replaced by a quasidisk

boundedly complete weak cauchy basic sequences in banach spaces with the pcp abstract it is proved that every non trivial weak cauchy sequence in a banach space with the pcp the point of continuity property has a boundedly complete basic subsequence the following result due independently to s bellenot and c finet is then deduced as a corollary if a banach space x has separable dual and the pcp then every non trivial weak cauchy sequence in x has a subsequence spanning an order one quasi reflexive space

close to convexity properties of gaussian hypergeometric functions abstract let a be the class of normalized analytic functions in the unit disk let z be either zf a b c z or c ab f a b c z 1 where f a b c z denotes the classical hypergeometric function the purpose of this paper is to study close to convexity and hence univalency of z in the unit disc more generally we find conditions on a b c and such that satisfies re e i 1 z z 0 for all z and for some real 1 2 1 2

conjugate operators that preserve the nonconvergence of bounded martingales we show that the conjugate t of an operator t x y with x and y banach spaces satisfies the following dichotomy either t preserves the nonconvergence of bounded martingales in y or there exists a compact operator k x y such that the kernel n t k fails the radon nikod m property

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

tukey classification of some ideals on ω and the lattices of weakly compact sets in banach spaces abstract we study the lattice structure of the family of weakly compact subsets of the unit ball b x of a separable banach space x equipped with the inclusion relation this structure is denoted by k b x and also with the parametrized family of almost inclusion relations k l e b x where e 0 this structure is denoted by ak b x tukey equivalence between partially ordered sets and a suitable extension to deal with ak b x are used assuming the axiom of analytic determinacy we prove that separable banach spaces fall into four categories namely k b x is equivalent either to a singleton or to or to the family k q of compact subsets of the rational numbers or to the family c of all finite subsets of the continuum also under the axiom of analytic determinacy a similar classification of ak b x is obtained for separable banach spaces not containing l 1 we prove in zfc that k b x ak b x are equivalent to either 0 k q or c the lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied

banach spaces with many boundedly complete basic sequences failing pcp we prove that there exist banach spaces not containing l1 failing the point of continuity property and satisfying that every semi normalized basic sequence has a boundedly complete basic subsequence this answers in the negative the problem of remark 2 in rosenthal 2007 12

remarks on generalized elliptic integrals we study the monotonicity for certain combinations of generalized elliptic integrals thus generalizing analogous well known results for classical complete elliptic integrals and prove a conjecture put forward by heikkala vamanamurthy and vuorinen