Log-convex Research Articles

Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios

Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios

Parabolic Frequency on Manifolds

Abstract We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are parabolic analogs of Almgren’s frequency function. When the static manifold is Euclidean space and the drift operator is the Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s frequency monotonicity for the ordinary heat equation. When the manifold is self-similarly evolving by the Ricci flow, we prove a parabolic frequency monotonicity for solutions of the heat equation. For the self-similarly evolving Gaussian soliton, this gives directly Poon’s monotonicity. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three-circle theorem about log convexity of holomorphic functions on C. From the monotonicity, we get parabolic unique continuation and backward uniqueness.

An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions

Abstract We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

Multi-parameter Mathieu, and alternating Mathieu series

The main purpose of this paper is to present a multi–parameter study of the familiar Mathieu series and the alternating Mathieu series S(r) and S˜(r). The computable series expansions of the their related integral representations are obtained in terms of higher transcendental hypergeometric functions like Lauricella’s hypergeometric function FC(m)[x], Fox–Wright Ψ function, Srivastava–Daoust S generalized Lauricella function, Riemann Zeta and Dirichlet Eta functions, while the extensions concern products of Bessel and modified Bessel functions of the first kind, hyper–Bessel and Bessel–Clifford functions. Auxiliary Laplace–Mellin transforms, bounding inequalities for the hyper–Bessel and Bessel–Clifford functions are established- which are also of independent but considerable interest. A set of bounding inequalities are presented either for the hyper–Bessel and Bessel–Clifford functions which are to our best knowledge new, or also for all considered extended Mathieu–type series. Next, functional bounding inequalities, log–convexity properties and Turán inequality results are presented for the investigated extensions of multi–parameter Mathieu–type series. We end the exposition by a thorough discussion closes the exposition including important details, bridges to occuring new questions like the similar kind multi–parameter treatment of the complete Butzer–Flocke–Hauss Ω function which is intimately connected with the Mathieu series family.

A ratio of finitely many gamma functions and its properties with applications

In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of finitely many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.

Domain Partition of the Hydro Production Function for Solving Efficiently the Short-Term Generation Scheduling Problem

Short-term generation scheduling (STGS) is a fundamental task in the operational planning analysis of hydroelectric plants. For the multi-unit case, the STGS is represented as a large-scale nonconvex mixed-integer nonlinear optimization model. Then, considering the (usual) short time for providing a solution, it is vital to exploit all the structural properties of the problem at hand. The main issue for exploiting this problem is the hydro production function (HPF), which is a nonlinear nonconvex relationship between power, head, and turbined outflow of a generating unit (GU). Nevertheless, the HPF usually presents operating regions where the function is convex and regions in which it is concave due to physical reasons. Inspired by sequential convex mixed-integer nonlinear programming techniques, this paper proposes partitioning the domain of the HPF in regions in which it is convex and those in which it is concave. The HPF is approximated by a piecewise linear function using the logarithmic aggregation convex combination (LACC) model in convex regions. In turn, in the concave regions, the HPF is replaced by a convex hull approach, which, combined with symmetry strategies, reduces the number of binary variables in the resulting optimization problem. Using several computational instances of a 50-unit hydroelectric plant, we show that the partitioning-based proposed strategy significantly reduces the computational time compared to two other efficient MILP formulations.

Strongly log-Convex Functions

Strongly log-Convex Functions

Uniqueness and exponential instability in a new two-temperature thermoelastic theory

<abstract> In this work we consider the temperature-rate dependent two temperatures thermoelastic theory. It has been proposed very recently. We study the case in which the elasticity tensor may not be positive definite. Thus, the problem can be ill posed in the sense of Hadamard. We adapt the logarithmic convexity argument to the specific situation proposed by this theory. That is, we define a suitable function on the solutions satisfying that the logarithm is convex. Uniqueness and instability of the solutions under suitable conditions on the constitutive tensors are proved. </abstract>

Continuous dependence for the Brinkman–Darcy–Kelvin–Voigt equations backward in time

We show that the solution to the Brinkman–Darcy–Kelvin–Voigt equations backward in time depends Hölder continuously upon the final data. A logarithmic convexity technique is employed, and uniqueness of the solution is simultaneously achieved.

Stability in Kelvin\u2013Voigt poroelasticity

Hölder continuous dependence of solutions upon the initial data is established for the linear theory of Kelvin–Voigt poroelasticity requiring only symmetry conditions upon the elastic coefficients. A novel functional is introduced to which a logarithmic convexity technique is employed.

APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many different applications, for example, boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the difference between exact and approximate solutions.

New Perspective of Log-Convex Functions

New Perspective of Log-Convex Functions

Some weighted trapezoidal inequalities for differentiable log-convex functions

Some new weighted trapezoidal inequalities for log-convex functions are established.

Unit-commitment via logarithmic aggregated convex combination in multi-unit hydro plants

We present a new strategy to solve the hydro unit commitment (HUC) in plants with many generating units (GUs), where the nonlinear Hydro Production Function (HPF) is linearized via Logarithmic Aggregated Convex Combination (LACC). Although the LACC is an efficient technique, the solution procedure via Mixed-Integer Linear Programming (MILP) solver in plants with several GUs is still a challenging task. Thus, this paper proposes a solution strategy that is divided into two steps. In the first step, the identical GUs are aggregated into an equivalent one, and the resulting MILP is resolved for obtaining the starting values of the gross head (GH). In the second step, the GH is used to determine the optimal dispatch considering the individualized HPF for each UG. The performance of the proposed strategy is analyzed in a Brazilian hydro plant composed of 50 GUs of two different types.

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

In this paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums. They establish three new inequalities involving finitely many exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers. They also survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finitely many gamma functions, present complete monotonicity of a linear combination of finitely many trigamma functions, construct a new ratio of finitely many gamma functions, derive monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finitely many gamma functions. Finally, they suggest two linear combinations of finitely many trigamma functions and two ratios of finitely many gamma functions to be investigated.

On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation

It is known that in the case that several constitutive tensors fail to be positive definite the system of the thermoelasticity could become unstable and, in certain cases, ill-posed in the sense of Hadamard. In this paper, we consider the Moore–Gibson–Thompson thermoelasticity in the case that some of the constitutive tensors fail to be positive and we will prove basic results concerning uniqueness and instability of solutions. We first consider the case of the heat conduction when dissipation condition holds, but some constitutive tensors can fail to be positive. In this case, we prove the uniqueness and instability by means of the logarithmic convexity argument. Second we study the thermoelastic system only assuming that the thermal conductivity tensor and the mass density are positive and we obtain the uniqueness of solutions by means of the Lagrange identities method. By the logarithmic convexity argument we prove later the instability of solutions whenever the elasticity tensor fails to be positive, but assuming that the conductivity rate is positive and the thermal dissipation condition hold. We also sketch similar results when conductivity rate and/or the thermal conductivity fail to be positive definite, but the elasticity tensor is positive definite and the dissipation condition holds. Last sections are devoted to considering the case when a third-order equation is proposed for the displacement (which comes from the viscoelasticiy). A similar study is sketched in these cases.

Hermite-Hadamard Type Inequalities for Composite Log-Convex Functions

© 2020 by authors, all rights reserved. Hermite-Hadamard type inequalities related to convex functions are widely being studied in functional analysis. Researchers have refined the convex functions as quasi-convex, h-convex, log-convex, m-convex, (α,m)-convex and many more. Subsequently, the Hermite-Hadamard type inequalities have been obtained for these refined convex functions. In this paper, we firstly review the Hermite-Hadamard type inequality for both convex functions and log-convex functions. Then, the definition of composite convex function and the Hermite-Hadamard type inequalities for composite convex functions are also reviewed. Motivated by these works, we then make some refinement to obtain the definition of composite log-convex functions, namely composite-ϕ−1 log-convex function. Some examples related to this definition such as GG-convexity and HG-convexity are given. We also define k-composite log-convexity and k-composite-ϕ−1 log-convexity. We then prove a lemma and obtain some Hermite-Hadamard type inequalities for composite log-convex functions. Two corollaries are also proved using the theorem obtained; the first one by applying the exponential function and the second one by applying the properties of k-composite log-convexity. Also, an application for GG-convex functions is given. In this application, we compare the inequalities obtained from this paper with the inequalities obtained in the previous studies. The inequalities can be applied in calculating geometric means in statistics and other fields.

Extremal determinants of Laplace\u2013Beltrami operators for rectangular tori

In this work we study the determinant of the Laplace–Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments. Refined logarithmic convexity and concavity results of the classical Jacobi theta functions of one real variable are deeply involved.

On Porous Matrices with Three Delay Times: A Study in Linear Thermoelasticity

Through the present work, we want to lay the foundation of the well-posedness question for a linear model of thermoelasticity here proposed, in which the presence of voids into the elastic matrix is taken into account following the Cowin–Nunziato theory, and whose thermal response obeys a three-phase lag time-differential heat transfer law. By virtue of the linearity of the model investigated, the basic initial-boundary value problem is conveniently modified into an auxiliary one; attention is paid to the uniqueness question, which is addressed through two alternative paths, i.e., the Lagrange identity and the logarithmic convexity methods, as well as to the continuous dependence issue. The results are achieved under very weak assumptions involving constitutive coefficients and delay times, at most coincident with those able to guarantee the thermodynamic consistency of the model.

Power series of one variable with condition of logarithmical convexity

We obtain a new version of the Hardy theorem about power series reciprocal to the power series with positive coefficients in this work. We prove that, if the sequence {an}, n ≥ K is logarithmically convex, then the reciprocal power series has only negative coefficients bn, n > 0 for any K if the first coefficient a0 is sufficiently large. The classical Hardy theorem corresponds to the case K = 0. These results are useful in the Nevanlinna–Pick theory. For instance, if the function k(x, y) can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{a}_{n}}{{{(x\bar {y})}}^{n}}} $$, an > 0, and the reciprocal function $$\frac{1}{{k(x,y)}}$$ can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{b}_{n}}{{{(x\bar {y})}}^{n}}} $$ such that bn 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc $$\mathbb{D}$$ with the Nevanlinna–Pick property. The reproducing kernel $$\frac{1}{{1 - x\bar {y}}}$$ of the classical Hardy space H2($$\mathbb{D}$$) is a prime example for our theorems. We also obtained new estimates on the growth of analytic functions reciprocal to analytic functions with positive Taylor coefficients.