further inequalities for the gamma function for x 0 and k 0 we present a method which permits us to obtain inequalities of the type k a x 1 r k x r k 1 k x 1 with the usual notation for the gamma function where a and are independent of k some examples are also given which improve well known inequalities finally we are also able to show in some cases that the values a and in the inequalities that we obtain cannot be improved

on gamma function inequalities on montre que certaines fonctions comportant les quotients de fonctions gamma sont completement monotones on en deduit des inegalites on etablit aussi la divisibilite infinie de plusieurs distributions de probabilite dont les transformees de laplace comportent des quotients de fonctions gamma

periodicity in a delayed ratio dependent predator prey system abstract with the help of a continuation theorem based on gaines and mawhin s coincidence degree easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio dependent predator prey system in a periodic environment

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

persistence and global stability for a delayed nonautonomous predator prey system without dominating instantaneous negative feedback abstract a delayed nonautonomous three species predator prey lotka volterra system without dominating instantaneous negative feedback is investigated it is proved that the system is uniformly persistent under appropriate conditions by constructing a suitable lyapunov functional sufficient conditions are derived for the global stability of the system

persistence in nonautonomous predator prey systems with infinite delays this paper studies the general nonautonomous predator prey lotka volterra systems with infinite delays the sufficient and necessary conditions of integrable form on the permanence and persistence of species are established a very interesting and important property of two species predator prey systems is discovered that is the permanence of species and the existence of a persistent solution are each other equivalent particularly for the periodic system with delays applying these results the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained some well known results on the nondelayed periodic predator prey lotka volterra systems are strongly improved and extended to the delayed case

existence and global stability of periodic solutions of a discrete predator prey system with delays with the help of continuation theorem in coincidence degree theory and lyapunov functions we derive sufficient and realistic conditions that guarantee the existence of a positive periodic solution for a delayed discrete predator prey system and also obtain sufficient conditions for the global stability of positive periodic solutions

kolmogorov landau inequalities for monotone functions where llfll ess su if t kolmogorov 2 suceeded in finding the best possible constants in 1 where the norm is as in 2 except for isolated results no new progress was made until schoenberg and cavaretta 3 found the best possible constants for llfll su if f we propose to prove inequalities about various clases of monotone functions which yield as special cases pointwise inequalities of the form 1 for example

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

some aspects of band limited signal extrapolation models discrete approximations and noise we present some theoretical results on the band limited signal extrapolation problem in section i we describe four basic models for the extrapolation problem these models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in 1 and 2 one of these models was shown to approximate the continuous band limited extrapolation problem 3 another model is obtained when the discrete fourier transform dft is used to implement the well known iterative algorithm given in 4 and 5 which was designed for solving the continuous extrapolation problem in section ii this model is related to the continuous model by means of an interesting approximation theorem also an important conjecture is presented section iii shows some approximation results specifically we prove that some discrete discrete and discrete continuous extrapolations of noisy signals converge to solutions of a certain continuous continuous noisy extrapolation problem when the noise is bounded by a known number max eta x leq epsilon this convergence is obtained by using normal families of entire functions in nand some other complex analysis tools we also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired this result indicates that in the presence of noise extrapolation techniques should be used judiciously in order to obtain reasonable results

asymptotic estimates for stieltjes constants a probabilistic approach let n n 0 be the sequence of stieltjes constants appearing in the laurent expansion of the riemann zeta function we obtain explicit upper bounds for n whose order of magnitude is as n tends to

a new extrapolation algorithm for band limited signals using the regularization method the ill posedness of the extrapolation problem in the presence of noise is considered a stable algorithm is constructed by solving a fredholm equation based on a regularization method the algorithm appears relatively robust since the noise eta sub delta t is taken as a function in l sup 2 t t t 0 such that the error energy integral mod eta sub delta t mod sup 2 dt

an efficient method for an ill posed problem band limited extrapolation by regularization in this paper a regularized spectral estimation formula and a regularized iterative algorithm for band limited extrapolation are presented the ill posedness is taken into account first the fredholm equation is regularized then it is transformed to a differential equation in the case where the time interval is r a fast algorithm to solve the differential equation is given by the finite difference and a regularized spectral estimation formula is obtained then a regularized iterative extrapolation algorithm is introduced and compared with the papoulis and gerchberg algorithm

some gamma function inequalities a class of completely monotonic functions are presented involving the gamma function as well as the derivative of the psi function as a consequence new upper and lower bounds for the ratio f x 1 f x s are obtained and compared with related bounds given in part by j d keckic and p m vasic our results are further applied to obtain functions which are laplace transforms of infinitely divisible probability measures

exponential gamma and polygamma functions simple proofs of classical and new inequalities abstract some inequalities for the exponential gamma and polygamma functions are established by means of the mean value theorem of differential calculus in particular the cases of the ratio of gamma and polygamma functions are considered in some cases the results are new in some others classical results are recovered or extended

on ramanujan s q function abstract this study provides a detailed analysis of a function which knuth discovered to play a central role in the analysis of hashing with linear probing the function named after knuth q n is related to several of ramanujan s investigations it surfaces in the analysis of a variety of algorithms and discrete probability problems including hashing the birthday paradox random mapping statistics the rho method for integer factorization union find algorithms optimum caching and the study of memory conflicts a process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the q n function in this way tight bounds are derived they answer a question of knuth the art of computer programming vol 1 1968 ex 1 2 11 3 13 itself a rephrasing of earlier questions of ramanujan in 1911 1913

inequalities for the median of the gamma distribution we provide several new inequalities involving l n the median of the gamma distribution of order n 1 with parameter 1 among others we present sharp upper and lower bounds for the arithmetic mean of l 1 l 2 l n for all integers n 1 we have an 76 n2 8405h nn email protected email protected k bn 76 n2 8405h nn with the best possible constants email protected k 1 l k k 23 8405k email protected email protected 1 683405 0 0080 here h n denotes the nth harmonic number

on a ramanujan equation connected with the median of the gamma distribution in this paper we consider the sequence n n 0 solving the ramanujan equation e n 2 n k 0 n k k n n n n 1 n 0 1 the three main achievements are the following we introduce a continuous time extension 0 t of on and show its close connections with the medians an of the n 1 1 distributions and the charlier polynomials we give upper and lower bounds for both 0 t and an in particular for on which are sharper than other known estimates finally we show and at the same time complete two conjectures by chen and rubin referring to the sequence of medians n n 1

signal extrapolation in wavelet subspaces the papoulis gerchberg pg algorithm is well known for band limited signal extrapolation the authors consider the generalization of the pg algorithm to signals in the wavelet subspaces in this research the uniqueness of the extrapolation for continuous time signals is examined and sufficient conditions on signals and wavelet bases for the generalized pg gpg algorithm to converge are given a discrete gpg algorithm is proposed for discrete time signal extrapolation and its convergence is investigated numerical examples are given to illustrate the performance of the discrete gpg algorithm

a fast convergence algorithm for band limited extrapolation by sampling in this paper an algorithm for band limited extrapolation is presented this uses shannon s sampling theorem and fourier series error analysis is given by proof the convergence rate is much faster than that of the papoulis gerchberg algorithm the procedure of the fast convergence algorithm is described in detail the result is demonstrated by numerical examples

estimates of generalized stieltjes constants with a quasi geometric rate of decay we approximate each generalized stieltjes constant n a by means of a finite sum involving bernoulli numbers such an approximation has a quasi geometric rate of convergence which improves as re a increases a more detailed analysis including numerical computations is carried out for the constants 1 and 1 1 the key point in the proof is a probabilistic representation of the aforementioned constants obtained as a consequence of a differential calculus concerning the gamma process

a sanz huang conjecture on band limited signal extrapolation with noise the authors prove that j sanz and t huang s ibid vol assp 33 p 1338 41 dec 1983 conjecture on bandlimited signal extrapolation with noise is indeed true all hypotheses and notations used in the proof are described in detail

periodic solution of a delayed predator prey system without dominating instantaneous negative feedback by using the continuation theorem base on gaines and mawhin s coincidence degree sufficient and realistic conditions are obtained for the global existence of positive periodic solution for a delayed predator prey system without dominating instantaneous negative feedback indeed our result are applicable to statedependent delays

interpolation and the discrete papoulis gerchberg algorithm analyze the performance of an iterative algorithm similar to the discrete papoulis gerchberg algorithm and which can be used to recover missing samples in finite length records of band limited data no assumptions are made regarding the distribution of the missing samples in contrast with the often studied extrapolation problem in which the known samples are grouped together indeed it is possible to regard the observed signal as a sampled version of the original one and to interpret the reconstruction result studied as a sampling result the authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data they give upper and lower bounds for the error as a function of the number of iterations together with the signals for which the bounds are attained also they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix from this analysis they infer the optimum value of the relaxation constant they also point out among all sampling sets with the same density those for which the convergence rate of the recovery algorithm is maximum or minimum for low pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer the worst convergence rates generally occur when the missing samples are contiguous

oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics abstract in this paper we shall consider the nonlinear delay differential equation where m and n are positive integers and v t and t are positive periodic functions of period in the nondelay case we shall show that has a unique positive periodic solution x t and provides sufficient conditions for the global attractivity of x t in the delay case we shall present sufficient conditions for the oscillation of all positive solutions of about x t and establish sufficient conditions for the global attractivity of x t

improved time censored reliability test plans for k out of n gamma systems abstract relevant prior knowledge supported by historical data and expert judgments is common in many manufacturing processes the classical viewpoint to the determination of acceptance sampling plans for k out of n gamma systems is generalized to those situations in which the available previous information is appreciable a beta distribution with a reduced parameter space is adopted to describe the stochastic behavior of the fraction of defective systems an integer nonlinear programming problem is stated to find the inspection scheme that minimizes the required number of components to test and keeps the average producer and consumer risks below specified thresholds first lower and upper bounds as well as an approximate solution are derived then an efficient step by step procedure which finds the best acceptance sampling plan in a finite number of iterations is provided the inclusion of prior information into the decision process provides substantial savings in the sample size required for demonstrating system reliability and also more accurate assessments of the actual producer and consumer risks a system of five water pumps for cooling a reactor is analyzed for illustrative and comparative purposes

observations on the extrapolation of a band limited signal problem in this correspondence the problem of extrapolating a time truncated band limited signal shall be considered with two objectives in mind 1 to respond to the sabri and steenaart 1 correspondence relative to the author s extrapolation method 2 and 2 to comment on the numerical difficulties which individuals have experienced when using various extrapolation procedures in the discussion that follows the terminology used in 2 shall be adopted

some aspects of band limited extrapolations in this paper some problems in band limited extrapolations are discussed these aspects include the computation of the inverse fourier transform the accuracy of the extrapolation the ill posedness and regularization method

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

completely monotonic functions associated with the gamma function and its q analogues several functions involving the gamma function x and the q gamma function q x are proved to be completely monotonic some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions

some inequalities and monotonicity properties associated with the gamma and psi functions abstract in this paper we present new completely monotonic functions and bernstein functions relating to the gamma and psi functions as consequences we establish new upper and lower bounds for the gamma function in terms of function also we present some estimates for stirling s formula remainder

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

logarithmically completely monotonic functions and applications in this article we give a sufficient condition and a necessary and sufficient condition for a function class associated with the gamma function to be logarithmically completely monotonic some applications are also presented

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

periodic solutions of periodic delay lotka volterra equations and systems abstract by using the continuation theorem of coincidence degree theory sufficient and realistic conditions are obtained for the existence of positive periodic solutions for both periodic lotka volterra equations and systems with distributed or state dependent delays our results substantially extend and improve existing results

a regularized sampling algorithm for reconstructing non bandlimited signals in this paper the reconstruction of non bandlimited sampling is discussed and a regularized sampling algorithm for non bandlimited signals is presented the error of the regularized sampling algorithm is presented and compared with the previous algorithm based on shannon s sampling theorem and compared with the tikhonov regularization method in the noisy case