Log-convex Research Articles

On the preservation of log convexity and log concavity under some classical Bernstein-type operators

This paper analyzes the preservation of both the log convexity and the log concavity under certain Bernstein-type operators. Some results are provided for the Bernstein, Szász, Baskakov, the gamma-type and the Weierstrass operators. Probabilistic methods support the proofs of these results.

Newton-type methods of high order and domains of semilocal and global convergence

We present the geometric construction of some classical iterative methods that have global convergence and “infinite” speed of convergence when they are applied to solve certain nonlinear equations f ( t ) = 0 . In particular, for nonlinear equations with the degree of logarithmic convexity of f ′ , L f ′ ( t ) = f ′ ( t ) f ‴ ( t ) / f ″ ( t ) 2 , is constant, a family of Newton-type iterative methods of high orders of convergence is constructed. We see that this family of iterations includes the classical iterative methods. The convergence of the family is studied in the real line and the complex plane, and domains of semilocal and global convergence are located.

A simple proof of logarithmic convexity of extended mean values

In the note, a simple proof is provided for the logarithmic convexity of extended mean values.

On The Hadamard's Inequality for Log-Convex Functions on the Coordinates

Inequalities of the Hadamard and Jensen types for coordinated log-convex functions defined in a rectangle from the plane and other related results are given.

On logarithmic convexity for differences of power means and related results

We give some further considerations about logarithmic convexity for differences of power Means for positive linear functionals as well as some related results. Mathematics subject classification (2000): 26A51, 26A46, 26A48.

Hardy's uncertainty principle, convexity and Schrödinger evolutions

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

A Calculus for Log-Convex Interference Functions

The behavior of certain interference-coupled multiuser systems can be modeled by means of logarithmically convex (log-convex) interference functions. In this paper, we show fundamental properties of this framework. A key observation is that any log-convex interference function can be expressed as an optimum over elementary log-convex interference functions. The results also contribute to a better understanding of certain quality-of-service (QoS) tradeoff regions, which can be expressed as sublevel sets of log-convex interference functions. We analyze the structure of the QoS region and provide conditions for the achievability of boundary points. The proposed framework of log-convex interference functions generalizes the classical linear interference model, which is closely connected with the theory of irreducible nonnegative matrices (Perron-Frobenius theory). We discuss some possible applications in robust communication, cooperative game theory, and max-min fairness.

The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: $T((a_n))_n=1/(a_0+\cdots +a_n)$. We determine the corresponding measure $\mu$, which has an increasing and convex density on $\mathopen]0,1\mathclose[$, and we study some analytic functions related to it. The Mellin transform $F$ of $\mu$ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on $\mathopen]-1,\infty\mathclose[$ satisfying $F(0)=1$ and the functional equation $1/F(s)=1/F(s+1)-F(s+1)$, $s>-1$.

On Logarithmic Convexity for Power Sums and Related Results

We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results.

On Logarithmic Convexity for Power Sums and Related Results II

In the paper "On logarithmic convexity for power sums and related results" (2008), we introduced means by using power sums and increasing function. In this paper, we will define new means of convex type in connection to power sums. Also we give integral analogs of new means.

On Logarithmic Convexity for Ky-Fan Inequality

We give an improvement and a reversion of the well-known Ky-Fan inequality as well as some related results.

The Euler equations from the point of view of differential inclusions

We show that for a Schrodinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity. One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrodinger equation on a product N × [0, T], where N is a closed manifold with a certain spectral gap. Examples of such N's are all (round) spheres n for n 1 and all Zoll surfaces. Finally, we discuss some examples arising in geometry of such manifolds and Schrodinger operators.

Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications

AbstractWe show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity.One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrödinger equation on a product N × [0, T], where N is a closed manifold with a certain spectral gap. Examples of such N's are all (round) spheres 𝕊n for n ≥ 1 and all Zoll surfaces.Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators.© 2007 Wiley Periodicals, Inc.

LOGARITHMIC CONVEXITY OF THE ONE-PARAMETER MEAN VALUES

In this article, the logarithmic convexity of the one-parameter mean values J(r) and the monotonicity of the product J(r)J(−r) with r ∈ R are presented. Some more general results are established.

On the Convexity of Feasible QoS Regions

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The feasible quality-of-service (QoS) region is the set of all QoS vectors that can be provided to the users by means of power control, with interference treated as noise. In an interference-limited scenario, this set is determined by the Perron root of some QoS-dependent nonnegative matrix. In a previous work, we showed that if the signal-to-interference ratio (SIR) is a log-convex function of the QoS, then the Perron root is a log-convex function. This implies convexity of the feasible QoS region. In this correspondence, we prove that the log-convexity property is also necessary for the Perron root to be convex for any choice of the (path) gain matrix. Interestingly, a significantly less restrictive property is sufficient when the gain matrix is confined to be symmetric positive semidefinite. </para>

On Logarithmic Convexity for Differences of Power Means

We proved a new and precise inequality between the differences of power means. As a consequence, an improvement of Jensen's inequality and a converse of Holder's inequality are obtained. Some applications in probability and information theory are also given.

Risk-sensitive capacity control in revenue management

Both the static and the dynamic single-leg revenue management problem are studied from the perspective of a risk-averse decision maker. Structural results well-known from the risk-neutral case are extended to the risk-averse case on the basis of an exponential utility function. In particular, using the closure properties of log-convex functions, it is shown that an optimal booking policy can be characterized by protection levels, depending on the actual booking class and the remaining time. Moreover, monotonicity of the protection levels with respect to the booking class and the remaining time are proven.

Inequalities involving modified Bessel functions of the first kind II

The intrinsic properties, including logarithmic convexity (concavity), of the modified Bessel functions of the first kind and some other related functions are obtained. Several inequalities involving functions under discussion are established.

Free vibrations of a polar body at elastic range

The purpose of this paper is to study certain features of the equations governing the time-harmonic free vibrations of a polar body at elastic range. The governing equations of micropolar elasticity are expressed in differential form, and then, the uniqueness of their solutions is investigated. The conditions sufficient for uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained that leads to all the governing equations of the free vibrations as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh’s quotient is given. As an application, the high-frequency vibrations of an elastic plate are treated.

Eigenvalue inequalities for convex and log-convex functions

We give a matrix version of the scalar inequality f( a + b) ⩽ f( a) + f( b) for positive concave functions f on [0, ∞). We show that Choi’s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic–geometric mean inequality for positive matrices is given. We also point out a simple proof of the Bhatia–Kittaneh arithmetic–geometric mean inequality.