completely monotonic degrees for a difference between the logarithmic and psi functions abstract in the paper the authors firstly present a concise proof for complete monotonicity of a function involving a difference between the logarithmic and psi functions secondly compute completely monotonic degree of the above mentioned function and finally pose several conjectures on completely monotonic degrees of remainders and their derivatives for the asymptotic formula of the logarithm of the classical euler gamma function

robust team decision theory this technical note considers linear quadratic team decision problems it shows that linear decisions are optimal and can be found by solving a linear matrix inequality

static team problems part i sufficient conditions and the exponential cost criterion the stationary conditions of radner are shown under relaxed hypotheses to be sufficient to establish the global optimality of candidate control laws for static team problems with convex cost this extension of radner s theorem establishes the optimality of affine laws for the exponential of a quadratic performance index with jointly gaussian state and observation variables the class of performance indexes of team problems for which the optimal solution is known is thereby enlarged

an inequality for ratios of gamma functions abstract let x denote euler s gamma function the following inequality is proved for y 0 and x 1 we have x y 1 y 1 1 x x y 2 y 1 1 x 1 x y x y 1 the inequality is reversed if 0 x 1 this resolves an open problem of guo and qi b n guo f qi inequalities and monotonicity for the ratio of gamma functions taiwanese j math 7 2003 239 247

most trees are short and fat this work proves new probability bounds relating to the height width and size of galton watson trees for example if t is any galton watson tree and h w and t are the height width and size of t respectively then h w has sub exponential tails and h t 1 2 has sub gaussian tails although our methods apply without any assumptions on the offspring distribution when information is provided about the distribution the method can be adapted accordingly and always seems to yield tight bounds

some completely monotonic functions involving polygamma functions and an application by using the first binet s formula the strictly completely monotonic properties of functions involving the psi and polygamma functions are obtained as direct consequences two inequalities are proved as an application the best lower and upper bounds of the nth harmonic number are established

inequalities asymptotic expansions and completely monotonic functions related to the gamma function abstract in this paper we present some completely monotonic functions and asymptotic expansions related to the gamma function based on the obtained expansions we provide new bounds for x 1 x 1 2 and x 1 2

remarks on some completely monotonic functions applying the euler maclaurin summation formula we obtain upper and lower polynomial bounds for the function xex 1 x 0 with coefficients the bernoulli numbers bk this enables us to give simpler proofs of some results of h alzer and f qi et al concerning complete monotonicity of certain functions involving the gamma function x the psi function x and the polygamma functions n x n 1 2

a class of strongly completely monotonic functions related to gamma function abstract a function f 0 r is called strongly completely monotonic if it has derivatives of all orders and 1 n x n 1 f n x is nonnegative and decreasing on 0 for all n 0 in this paper we prove the function g n x ln x 1 2 x ln x x 1 2 ln 2 k 1 n 1 2 1 2 k b 2 k 2 k 2 k 1 x 2 k 1 is strongly completely monotonic on 0 using the same technique we give an alternative proof of a known result moreover two conjectures are proposed

inequalities and asymptotic expansions associated with the ramanujan and nemes formulas for the gamma function we present some inequalities and asymptotic expansions associated with the ramanujan and nemes formulas for the gamma function

approximations for certain hyperbolic functions by partial sums of their taylor series and completely monotonic functions related to gamma function abstract in this paper we establish some lower and upper bounds for certain hyperbolic functions in terms of partial sums of their taylor series these allow us to present two completely monotonic functions involving gamma function as consequences sharp burnside type approximations for gamma function sharp detemple wang type approximations for psi function and sharp bounds for polygamma functions are deduced which generalize and improve some known results

monotonicity of some functions involving the gamma and psi functions let l x x x t x s x 8 t 1 where x is euler s gamma function we determine conditions for the numbers s t so that the function x x s x t x t s 1 l x is strongly completely monotonic on 0 oo through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation we also give several other examples of strongly completely monotonic functions defined in terms of and functions some limiting and particular cases are also considered

monotonicity properties inequalities and asymptotic expansions associated with the gamma function we define x by the equality x 1 x e x 8 x 3 4 x 2 x 1 30 1 6 e x we call x the remainder of ramanujan s formula in this paper we present some properties for x including monotonicity properties inequalities and asymptotic expansions furthermore we present some full asymptotic expansions for the gamma function related to the nemes ramanujan and burnside formulas

centralized and decentralized solutions of the linear exponential gaussian problem a particular class of stochastic control problems constrained to different information patterns is considered this class consists of minimizing the expectation of an exponential cost criterion with quadratic argument subject to a discrete time gauss markov dynamic system i e the linear exponential gaussian leg control problem besides the one step delayed information pattern previously considered the classical and the one step delayed information sharing osdis patterns are assumed after determining the centralized controller based upon the classical information pattern the optimal decentralized controller based upon the osdis pattern and the solution to a static team problem is found to be affine a unifying approach to determine controllers based upon these three information patterns is obtained by noting that the value of a quadratic exponent of an exponential function is independent of the information structure both necessary and sufficient conditions for the controllers to be minimizing are obtained regardless of the exponential form the negative exponential form is included which is unimodal but not convex

an infinite class of completely monotonic functions involving the q gamma function abstract in this paper the q analogue of the stirling formula for the q gamma function is used to prove the complete monotonicity property for an infinite class of functions which are all closely related to the q gamma function and its logarithmic derivatives q polygamma functions as an application of this result the two sided inequalities for the q gamma and the q polygamma functions are established

completely monotonic functions of positive order and asymptotic expansions of the logarithm of barnes double gamma function and euler s gamma function abstract we introduce completely monotonic functions of order r 0 and show that the remainders in asymptotic expansions of the logarithm of barnes double gamma function and euler s gamma function give rise to completely monotonic functions of any positive integer order

sub gaussian tail bounds for the width and height of conditioned galton watson trees we study the height and width of a galton watson tree with offspring distribution satisfying e 1 0 var conditioned on having exactly n nodes under this conditioning we derive sub gaussian tail bounds for both the width largest number of nodes in any level and height greatest level containing a node the bounds are optimal up to constant factors in the exponent under the same conditioning we also derive essentially optimal upper tail bounds for the number of nodes at level k for 1 k n

on ruijsenaars asymptotic expansion of the logarithm of the double gamma function abstract we study the remainder r n x in an asymptotic expansion due to s n m ruijsenaars for the logarithm of the double gamma function we show that for any even number n the function 1 n 2 1 r n x is completely monotonic on 0 this proves a recent conjecture of h l pedersen in addition we give an estimate for the remainder r n x for n even

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

sub exponential tail bounds for conditioned stable bienayme galton watson trees we establish uniform sub exponential tail bounds for the width height and maximal outdegree of critical bienayme galton watson trees conditioned on having a large fixed size whose offspring distribution belongs to the domain of attraction of a stable law this extends results obtained for the height and width by addario berry devroye and janson in the finite variance case

conceptual proofs of l log l criteria for mean behavior of branching processes the kesten stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process showing that an l log l condition is decisive in critical and subcritical cases results of kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations we give conceptual proofs of these theorems based on comparisons of galton watson measure to another measure on the space of trees this approach also explains yaglom s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution

some inequalities for the trigamma function in terms of the digamma function in the paper the authors establish three kinds of inequalities for the trigamma function in terms of the exponential function to powers of the digamma function these newly established inequalities extend some known results the method in the paper utilizes some facts from the asymptotic theory and is a natural way to solve problems for approximating some quantities for large values of the variable

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