asymptotic behaviour of the bessel heat kernels we consider dirichlet heat kernel p a mu t x y for the bessel differential operator l mu frac d 2 dx 2 frac 2 mu 1 2x mu in mathbf r in half line a infty a 0 and provide its asymptotic expansions for xy t rightarrow infty

the equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

the bochner riesz means for fourier bessel expansions abstract the bochner riesz means for fourier bessel expansions are analyzed we prove a uniform two weight inequality with potential weights for these means the result provides necessary and sufficient conditions for boundedness moreover we obtain some corollaries regarding the convergence of these means and the boundedness of other operators related to the fourier bessel series

s asymptotically ω periodic and asymptotically ω periodic solutions to semi linear cauchy problems with non dense domain abstract in this work we study the existence and uniqueness of s asymptotically periodic and asymptotically periodic solutions to a first order differential equation with linear part dominated by a hille yosida operator with non dense domain applications to partial differential equations fractional integro differential and neutral differential equations are given

intrinsic ultracontractivity and the dirichlet laplacian we show that if d x y 0 x 1 f x y 1 where f is negative upper semicontinuous and bounded below then the semigroup associated with the dirichlet laplacian on d is intrinsically ultracontractive analogs in higher dimensions are proved we construct domains of infinite area with this property

a fatou theorem for the solution of the heat equation at the corner points of a cylinder in this paper the author proves existence and uniqueness of the initial dirichlet problem for the heat equation in a cylindrical domain d 0 where d is a bounded smooth domain in r n with zero lateral values a unique representation of the strong solution is given in terms of measures on d and on d we also show that the strong solution u x t of the heat equation in a cylinder converges a e x 0 d 0 as x t converges to points on d 0 along certain nontangential paths

solving nonlinear matrix equations arising in tree like stochastic processes abstract in this paper based on matrix structure analysis we derive and analyze efficient algorithms to solve nonlinear matrix equations of the form x 1 i d a i x 1 d i c this class of equations is encountered in the solution of tree like stochastic processes which are a generalization of quasi birth and death qbd processes

on stepanov almost periodic semigroups and cosine functions of operators abstract in this paper we consider stepanov asymptotically almost periodic sp a a p mild solutions of first and second order abstract differential equations specifically we prove that a c0 semigroup is sp a a p if and only if it is asymptotically almost periodic and that a c0 cosine function of operators is sp a a p if and only if it is almost periodic

newton s iteration for the extinction probability of a markovian binary tree abstract continuous time multi type branching processes have applications in a large number of fields such as biology and telecommunication systems a basic problem for this kind of processes is to determine the extinction probability and in order to compute it it is necessary to find the minimal nonnegative solution of a non linear matrix equation we consider here a particular family of branching processes called markovian binary trees these give rise to second order equations and we apply newton s method for fixed point equations we show that this algorithm is well defined and converges quadratically in the domain of interest we also give it a probabilistic interpretation in terms of the branching process itself

littlewood paley stein gk functions for fourier bessel expansions abstract g k functions related to the poisson semigroup of fourier bessel expansions are defined for each k 1 it is proved that these g k functions are calderon zygmund operators in the sense of the associated space of homogeneous type hence their mapping properties follow from the general theory

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

sharp estimates for the green function in lipschitz domains we give sharp global estimates for the green function and martin kernel in lipschitz domains we apply the estimates to obtain short proofs of some classical results

potential operators associated with jacobi and fourier bessel expansions abstract we study potential operators riesz and bessel potentials associated with classical jacobi and fourier bessel expansions we prove sharp estimates for the corresponding potential kernels then we characterize those 1 p q for which the potential operators are of strong type p q of weak type p q and of restricted weak type p q these results may be thought of as analogues of the celebrated hardy littlewood sobolev fractional integration theorem in the jacobi and fourier bessel settings as an ingredient of our line of reasoning we also obtain sharp estimates of the poisson kernel related to fourier bessel expansions

jacobi transplantation revisited a transplantation theorem for jacobi series proved by muckenhoupt is reinvestigated by means of a suitable variant of calderon zygmund operator theory an essential novelty of our paper is weak type 1 1 estimate for the jacobi transplantation operator located in a fairly general weighted setting moreover l p estimates are proved for a class of weights that contains the class admitted in muckenhoupt s theorem

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

asymptotically periodic solutions of abstract differential equations abstract in this paper we introduce the concept of s asymptotically periodicity in the stepanov sense we study some properties of the space consisting of s asymptotically periodic functions and we use this notion to establish the existence of s asymptotically periodic solutions to linear and semi linear first order abstract differential equations in particular we characterize the strongly continuous semigroups of bounded linear operators defined on reflexive spaces which are s asymptotically periodic as those semigroups which are the direct sum of an periodic and a strongly stable semigroup

convergence rates of some iterative methods for nonsymmetric algebraic riccati equations arising in transport theory we determine and compare the convergence rates of various fixed point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic riccati equations arising in transport theory

inequalities for bessel functions we derive some inequalities for the cylinder function c x defined by cv x jv x cos yv x sin 0 where j x and y x denote the bessel functions of the first and second kind respectively our chief tools are three recurrence relations for c x and an integral formula for j x

inequalities for some special functions abstract inequalities for bessel functions modified bessel functions and of their ratios are obtained these results are either sharper or hold under weaker conditions than had been known earlier similar inequalities are proved for the function w v t ti v t i v 1 t which is useful in some problems of finite elasticity in the sequel inequalities for confluent hypergeometric functions and of their ratios are established

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

mean and weak convergence of fourier bessel series abstract we study the uniform boundedness on some weighted lp spaces of the partial sum operators associated to fourier bessel series obtaining necessary and sufficient conditions for this boundedness in terms of the weights on the other hand we study the weak and restricted weak type of the partial sum operators in the end points of the interval of strong boundedness

fourier bessel heat kernel estimates we provide sharp two sided estimates of the fourier bessel heat kernel and we give sharp two sided estimates of the transition probability density for the bessel process in 0 1 killed at 1 and killed or reflected at 0

bounds for modified bessel functions abstract in this paper we derive inequalities for the modified bessel functions i v t and k v t in some cases our results are completely new in other cases we improve or extend known results similar inequalities are proved for the function w v ti v t i v 1 t which is useful in some problems of finite elasticity

transplantation and multiplier theorems for fourier bessel expansions proved are weighted transplantation inequalities for fourier bessel expansions these extend known results on this subject by considering the largest possible range of parameters allowing more weights and admitting a shift the results are then used to produce a fairly general multiplier theorem with power weights for considered expansions also fractional integral results and conjugate function norm inequalities for these expansions are proved

perturbation analysis of the extinction probability of a markovian binary tree abstract the extinction probability of the markovian binary tree mbt is the minimal nonnegative solution of a quadratic vector equation qve in this paper we present a perturbation analysis for the extinction probability of a supercritical mbt we derive a perturbation bound for the minimal nonnegative solution of the qve which is a bound on the difference between the solutions of two nearby equations in terms of the perturbation magnitude a posteriori error bound is also given which is a bound on the distance between an approximate solution and the real solution in terms of the residual of the approximate solution numerical experiments show that these bounds are fairly sharp

landauʼs necessary density conditions for the hankel transform abstract we will prove an analogue of landau s necessary conditions h j landau necessary density conditions for sampling and interpolation of certain entire functions acta math 117 1967 37 52 for spaces of functions whose hankel transform is supported in a measurable subset s of the positive semi axis as a special case necessary density conditions for the existence of fourier bessel frames are obtained

two weight norm inequalities for the cesaro means of hermite expansions an accurate estimate is obtained of the cesaro kernel for hermite expansions this is used to prove two weight norm inequalities for cesaro means of hermite polynomial series and for the supremum of these means these extend known norm inequalities even in the single power weight and unweighted cases an almost everywhere convergence result is obtained as a corollary it is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the cesaro supremum of the means are necessary

on the solution of algebraic riccati equations arising in fluid queues abstract new algorithms for solving algebraic riccati equations are which arise in fluid queues models are introduced they are based on reducing the are to a unilateral quadratic matrix equation of the kind ax 2 bx c 0 and on applying the cayley transform in order to arrive at a suitable spectral splitting of the associated matrix polynomial a shifting technique for removing unwanted eigenvalues of modulus 1 is complemented with a suitable parametrization of the matrix equation in order to arrive at fast and numerically reliable solvers based on quadratically convergent iterations like logarithmic reduction and cyclic reduction numerical experiments confirm the very good performance of these algorithms

turan type inequalities for kratzel functions abstract complete monotonicity laguerre and turan type inequalities are established for the so called kratzel function z defined by z u 0 t 1 e t u t d t where u 0 and r moreover we prove the complete monotonicity of a determinant function of which entries involve the kratzel function

harmonic analysis operators associated with multidimensional bessel operators we establish that the maximal operator and the littlewood paley g function associated with the heat semigroup defined by multidimensional bessel operators are of weak type 1 1 we also prove that riesz transforms in the multidimensional bessel setting are of strong type p p for every 1 p

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

maximal function characterizations for hardy spaces on spaces of homogeneous type with finite measure and applications abstract we prove nontangential and radial maximal function characterizations for hardy spaces associated to a non negative self adjoint operator satisfying gaussian estimates on a space of homogeneous type with finite measure this not only addresses an open point in the literature but also gives a complete answer to the question posed by coifman and weiss in the case of finite measure we then apply our results to give maximal function characterizations for hardy spaces associated to second order elliptic operators with neumann and dirichlet boundary conditions schrodinger operators with dirichlet boundary conditions and fourier bessel operators

quadratic vector equations in this chapter we aim to study in an unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses specific cases of these problems have been studied extensively in the past by several authors for references to the single equations and results we refer the reader to the following sections in particular section 2 3 many of the results appearing here have already been proved for one or more of the single instances of the problems resorting to specific characteristics of the problem in some cases the proofs we present here are mere rewritings of the original proofs with a little change of notation to adapt them to our framework but in some cases we are effectively able to remove some hypotheses and generalize the results by abstracting the specific aspects of each problem

existence of s asymptotically ω periodic solutions for fractional order functional integro differential equations with infinite delay abstract we study s asymptotically periodic solutions of the abstract fractional equation u 1 a u f t u t 1 2 where a is a linear operator of sectorial type 0

estimates on green functions and schrodinger type equations for non symmetric diffusions with measure valued drifts abstract in this paper we establish sharp two sided estimates for the green functions of non symmetric diffusions with measure valued drifts in bounded lipschitz domains as consequences of these estimates we get a 3g type theorem and a conditional gauge theorem for these diffusions in bounded lipschitz domains informally the schrodinger type operators we consider are of the form l where l is a uniformly elliptic second order differential operator is a vector valued signed measure belonging to k d 1 and is a signed measure belonging to k d 2 in this paper we establish two sided estimates for the heat kernels of schrodinger type operators in bounded c 1 1 domains and a scale invariant boundary harnack principle for the positive harmonic functions with respect to schrodinger type operators in bounded lipschitz domains

two weight norm inequalities for the cesaro means of generalized hermite expansions we prove two weight norm inequalities for cesaro means of generalized hermite polynomial series and for the supremum of these means a result about weak boundedness and an almost everywhere convergence result are also obtained

fast iterative schemes for nonsymmetric algebraic riccati equations arising from transport theory for a class of nonsymmetric algebraic riccati equations arising from transport theory lu siam j matrix anal appl 26 2005 pp 679 685 reformulated them into a couple of fixed point equations in a vector form and constructed an iterative method for finding their minimal positive solutions in this paper based on this fixed point equation we propose several fast and effective iterative methods for solving the above mentioned special type of nonsymmetric algebraic riccati equations these iterative methods can be categorized into the class of nonlinear block relaxation iterations and can also be considered as accelerated variants of the known fixed point iteration according to the principle of using the currently available information as promptly as possible the monotone convergence theorems about these new iterative methods are established and the sharper bounds about the solutions of the nonsymmetric algebraic riccati equations are derived numerical implementations show that the new iterative methods are feasible and effective solvers for the nonsymmetric algebraic riccati equations arising from transport theory

weak type estimates for bochner riesz spherical summation multipliers we consider the bochner riesz multiplier tofw 1 1 12 tjw 8 0 where denotes the fourier transform it is shown that the multiplier operator to is weak type po po acting on lpo rn radial functions where po is the critical value 2n n 1 28

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

two weight norm inequalities for cesaro means of laguerre expansions two weight lp norm inequalities are proved for cesaro means of laguerre polynomial series and for the supremum of these means these extend known norm inequalities even in the single power weight and unweighted cases by including all values of p 1 for all positive orders of the cesaro summation and all values of the laguerre parameter a 1 almost everywhere convergence results are obtained as a corollary for the cesaro means the hypothesized conditions are shown to be necessary for the norm inequalities necessity results are also obtained for the norm inequalities with the supremum of the cesaro means in particular for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth

bounds for modified struve functions of the first kind and their ratios abstract we obtain a simple two sided inequality for the ratio l x l 1 x in terms of the ratio i x i 1 x where l x is the modified struve function of the first kind and i x is the modified bessel function of the first kind this result allows one to use the extensive literature on bounds for i x i 1 x to immediately deduce bounds for l x l 1 x we note some consequences and obtain further bounds for l x l 1 x by adapting techniques used to bound the ratio i x i 1 x we apply these results to obtain new bounds for the condition numbers x l x l x the ratio l x l y and the modified struve function l x itself amongst other results we obtain two sided inequalities for x l x l x and l x l y that are given in terms of x i x i x and i x i y respectively which again allows one to exploit the substantial literature on bounds for these quantities the results obtained in this paper complement and improve existing bounds in the literature

global existence and stability of solutions of matrix riccati equations we consider a matrix riccati equation containing two parameters c and the quantity c denotes the average total number of particles emerging from a collision which is assumed to be conservative i e 0 c 1 and 0 1 is an angular shift let s c 0 c 1 and 0 1 stability analysis for two steady state solutions xmin and xmax are provided in particular we prove that xmin is locally asymptotically stable for s 1 0 while xmax is unstable for s 1 0 for c 1 and 0 xmin xmax is neutral stable we also show that such equations have a global positive solution for c s provided that the initial value is small and positive

on s asymptotically ω periodic functions on banach spaces and applications abstract this paper is devoted to the study of the class of continuous and bounded functions f 0 x for which exists 0 such that lim t f t f t 0 in the sequel called s asymptotically periodic functions we discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically periodic functions we also study the existence of s asymptotically periodic mild solutions of the first order abstract cauchy problem in banach spaces

new bounds for the generalized marcum q function in this paper we study the generalized marcum q function q nu a b where a nu 0 and b ges 0 our aim is to extend the results of corazza and ferrari ieee trans inf theory vol 48 pp 3003 3008 2002 to the generalized marcum q function in order to deduce some new tight lower and upper bounds the key tools in our proofs are some monotonicity properties of certain functions involving the modified bessel function of the first kind and some classical inequalities i e the cauchy buniakowski schwarz and chebyshev integral inequalities these bounds are shown to be very tight for large b i e the relative errors of our bounds converge to zero as b increases both theoretical analysis and numerical results are provided to show the tightness of our bounds

efficient methods for solving a nonsymmetric algebraic riccati equation arising in stochastic fluid models we consider the nonsymmetric algebraic riccati equation xm12x xm11 m22x m21 0 where m11 m12 m21 m22 are real matrices of sizes n n n m m n m m respectively and m mij 2i j 1 is an irreducible singular m matrix with zero row sums the equation plays an important role in the study of stochastic fluid models where the matrix m is the generator of a markov chain the solution of practical interest is the minimal nonnegative solution this solution may be found by basic fixed point iterations newton s method and the schur method however these methods run into difficulties in certain situations in this paper we provide two efficient methods that are able to find the solution with high accuracy even for these difficult situations

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

weighted sp pseudo s asymptotic periodicity and applications to volterra integral equations abstract this paper is related to the function space formed by weighted sp pseudo s asymptotic periodicity and their applications initially the translation invariance and completeness of the function space are investigated additionally the composition theorem and convolution operator generated by lebesgue integrable functions are presented finally existence and uniqueness of solutions with weighted sp pseudo s asymptotic periodicity for two classes of volterra equations are proved by using the results obtained above and some concrete examples are given the methods mainly include minkowski s inequality convolution inequality contraction mapping principle and especially the generalized minkowski s inequality our results extend some known results on asymptotic periodicity

on the link between markovian trees and tree structured markov chains abstract in this paper we describe a link between markovian binary trees mbt and tree like quasi birth and death processes tlqbd by associating a specific tlqbd to each mbt the algorithms to compute the matrices g k in the tlqbd then correspond to the algorithms calculating the extinction probability vector of the mbt this parallelism leads to a new quadratic algorithm based on the newton iteration method which converges to the extinction probability of an mbt we also present a one to one correspondence between a general markovian tree gmt and a scalar tree structured m g 1 type markov chain this allows us to prove the equivalence between the main result on the positive recurrence null recurrence or transience of a scalar tree structured m g 1 type markov chain and the criticality of a gmt

ml detection in phase noise impaired simo channels with uplink training the problem of maximum likelihood ml detection in training assisted single input multiple output simo systems with phase noise impairments is studied for two different scenarios i e the case when the channel is deterministic and known constant channel and the case when the channel is stochastic and unknown fading channel furthermore two different operations with respect to the phase noise sources are considered namely the case of identical phase noise sources and the case of independent phase noise sources over the antennas in all scenarios the optimal detector is derived for a very general parameterization of the phase noise distribution furthermore a high signal to noise ratio snr analysis is performed to show that symbol error rate ser floors appear in all cases the ser floor in the case of identical phase noise sources for both constant and fading channels is independent of the number of antenna elements in contrast the ser floor in the case of independent phase noise sources is reduced when increasing the number of antenna elements for both constant and fading channels finally the system model is extended to multiple data channel uses and it is shown that the conclusion is valid for these setups as well

necessity conditions for multipliers with power weights it is shown that if multiplier operators are bounded on lp with weight xla for all functions in the space s 0o of schwartz functions whose founer transforms have compact support not including 0 and all multiplier functions in a standard tiormander type multiplier class then oz must satisfy certain inequalities this is a sequel to a previous paper in which conditions on a that were almost the same were shown to be sufficient for the norm irlequality to hold

asymptotically almost periodic solutions of abstract differential equations abstract in this paper we are concerned with the problem of existence and uniqueness of asymptotically almost periodic mild solutions of nonhomogeneous first order abstract differential equations essentially we obtain existence of asymptotically almost periodic solutions when the semigroup is asymptotically almost periodic or when the forcing term is an asymptotically stepanov almost periodic function

a note on the minimal nonnegative solution of a nonsymmetric algebraic riccati equation for the nonsymmetric algebraic riccati equation for which the four coefficient matrices form an m matrix the solution of practical interest is often the minimal nonnegative solution in this note we prove that the minimal nonnegative solution is positive when the m matrix is irreducible

chasing volatility a persistent multiplicative error model with jumps abstract persistence and unpredictable large increments characterize the volatility of financial returns we propose the multiplicative error model with volatility jumps mem j to describe and predict the probability and the size of these extreme events under the mem j the conditional density of the realized measure is a countably infinite mixture of gamma and kappa distributions with closed form conditional moments we derive stationarity conditions and the asymptotic theory for the maximum likelihood estimation estimates of the volatility jump component confirm that the probability of jumps dramatically increases during the financial crises the mem j improves over other models with fat tails

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

on quasi periodic properties of fractional sums and fractional differences of periodic functions this article is devoted to the study of discrete fractional calculus our goal is to investigate quasi periodic properties of fractional order sums and differences of periodic functions using riemann liouville and caputo type definitions we study concepts close to the well known idea of periodic function such as asymptotically periodicity or s asymptotically periodicity we use basic tools of discrete fractional calculus boundedness of sums and differences of fractional order of a given periodic function is also investigated