Nevanlinna's lemma on the logarithmic derivative of a meromorphic function

In proving the second fundamental theorem for functions meromorphic in the half-plane, Nevanlinna has asserted (in 1925) that they satisfy a lemma similar to the well-known lemma on the logarithmic derivative, but his proof was based on additional assumptions. These assumptions were later relaxed by Dufresnoy (in 1939) and Ostrovskii (in 1961). Here we shall show that in the general case, functions which are meromorphic in the half-plane do not satisfy a lemma similar to the lemma on the logarithmic derivative.     

Note on a Continued Fraction of Ramanujan

Ramanujan, who loved continued fractions, recorded many of his formulas in his two notebooks. The 25th entry in Chapter 12 of Notebook II is a continued fraction involving a quotient of gamma function products. We are going to give a new proof of Entry 25, which, incidently, is the least difficult of all Ramanujan's continued fraction formulas involving quotients of gamma function products. In our proof, we make use of a hypergeometric formula of his.     

Quasi-isometric classification of non-uniform lattices in semisimple groups of higher rank

We give another proof of the quasi-isometric classification theorem of non-uniform lattices in higher rank semisimple groups. We use the asymptotic cone and a class of ats (the logarithmic ats) which move away in the cusp with logarithmic speed. Submitted: October 1998, Revised version: July 1999, Final version: October 1999.     

Local Remez-type inequalities for exponentials of a potential on a piecewise analytic ARC

Based on techniques from potential theory and geometric function theory, we give a new proof of the pointwise Remez-type inequality for exponentials of logarithmic potentials on the unit interval [−1, 1]. This inequality was first proved by Erdélyi, Li and Saff [8, Theorem 2.2.]. We generalize our result to the case of a piecewise analytic arc.     

From concentration to logarithmic Sobolev and Poincaré inequalities

We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand's transport inequality.     

A remark on a logarithmic functional equation

We revisit the logarithmic functional equation of Heuvers and Kannappan [K.J. Heuvers, Pl. Kannappan, A third logarithmic functional equation and Pexider generalizations, Aequationes Math. 70 (2005) 117-121] and give a simple proof of the result and discuss the locally integrable solutions of the equation.     

Complexity penalized support estimation

We consider the estimation of the support of a probability density function with iid observations. The estimator to be considered is a minimizer of a complexity penalized excess mass criterion. We present a fast algorithm for the construction of the estimator. The estimator is able to estimate supports which consists of disconnected regions. We will prove that the estimator achieves minimax rates of convergence up to a logarithmic factor simultaneously over a scale of Hölder smoothness classes for the boundary of the support. The proof assumes a sharp boundary for the support.     

Partitions and functional Santaló inequalities

We give a direct proof of a functional Santaló inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santaló inequality. The argument combines a logarithmic form of the Prékopa-Leindler inequality and a partition theorem of Yao and Yao. Received: 21 August 2008     

Optimal ellipsoidal approximations around the Analytic center

We present a simple and self-contained proof for two-sided ellipsoidal approximations of certain convex setsS. The ellipsoids are centered at the minimum of the logarithmic barrier function forS. The ratio of inner and outer ellipsoid is optimal with respect to a self-concordance parameter.     

Nondegenerate saddle points and the absence of mixing in flows on surfaces

A detailed proof of the absence of mixing is presented for a special flow constructed by an arbitrary rotation of the circle and by a symmetric function with logarithmic singularities (i.e., a function for which the sums of the coefficients of logarithms for “right” and “left” singularities are equal).     

Sharp weighted logarithmic bound for maximal operators

The paper contains the study of sharp weighted logarithmic estimates for maximal operators on probability spaces equipped with a tree-like structure. These inequalities can be regarded as LlogL versions of the classical estimates of Fefferman and Stein. The proof exploits the existence of a certain special function, enjoying appropriate majorization and concavity conditions.     

Stability of the determination of the surface impedance of an obstacle from the scattering amplitude

We prove a stability estimate of logarithmic type for the inverse problem consisting in the determination of the surface impedance of an obstacle from the scattering amplitude. We present a simple and direct proof, which is essentially based on an elliptic Carleman inequality. Copyright © 2013 John Wiley & Sons, Ltd.     

Some properties of the Lozinskii logarithmic norm

We study properties of the logarithmic matrix norm. We obtain new estimates for matrix norms as well as for the spectral radius and the spectral abscissa. We give a new proof of the Fiedler theorem and a block test for the Hurwitz property of a matrix based on the theory of nonnegative and off-diagonal-nonnegative matrices.     

Discrete Morse theory and the consecutive pattern poset

We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrímsson’s formula for the M?bius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier, Bj?rner determined the M?bius function and homotopy type of factor order and the results are remarkably similar to those in the pattern case. In his thesis, Willenbring used discrete Morse theory to give an illuminating proof of Bj?rner’s result. Since our proof parallels Willenbring’s, we also consider the relationship between the two posets. In particular, we show that some of their intervals are isomorphic, and also that there is a sequence of posets interpolating between the two all of whom have essentially the same M?bius function.     

Harmonic and Superharmonic Majorants on the Disk

A proof is given to show that a positive function on the unitdisk admits a harmonic majorant if and only if it has a certainexplicit upper envelope that admits a superharmonic majorant.The (logarithmic) Lipschitz regularity of this superharmonicmajorant is discussed. 2000 Mathematics Subject Classification31A05.     

Linear Differential Equations and Logarithmic Derivative Estimates

We prove two sharp inequalities for the growth of solutionsof certain linear differential equations in the unit disk. Forthe proofs of these inequalities, we use the method of successiveapproximations and sharp estimates for the logarithmic derivativesof finite order meromorphic functions in the unit disk. Thesetechniques can also be used to give an alternate proof of awell-known result in the plane. The sharp logarithmic derivativeestimates are a corollary of general estimates, and all theseestimates have independent interest. 2000 Mathematics SubjectClassification 34M10, 30D35.     

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

对数凸模糊映射

利用模糊数的表示定理,本文证明了对数凸模糊映射一些基本性质,并纠正了S.N anda和K.K ar对数凸模糊映射定义的不合理性和其证明中的错误。     

The Rogers-Ramanujan continued fraction and a new Eisenstein series identity

With two elementary trigonometric sums and the Jacobi theta function θ₁, we provide a new proof of two Ramanujan's identities for the Rogers-Ramanujan continued fraction in his lost notebook. We further derive a new Eisenstein series identity associated with the Rogers-Ramanujan continued fraction.     

Optimal impulse control of a portfolio with a fixed transaction cost

The aim of this work is to investigate a portfolio optimization problem in presence of fixed transaction costs. We consider an economy with two assets: one risky, modeled by a geometric Brownian motion, and one risk-free which grows at a certain fixed rate. The agent is fully described by his/her utility function and the objective is to maximize the expected utility from the liquidation of wealth at a terminal date. We deal with different forms of utility functions (power, logarithmic and exponential utility), describing in each case how the fixed transaction costs influence the agent’s behavior. We show when it is optimal to recalibrate his/her portfolio and which are the best adjusted portfolios. We also analyze how the optimal strategy is influenced by the risk-aversion, as well as other model parameters.