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About 72 papers for “Logarithmically convex function”

Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions

Tie‐Hong Zhao, Yu‐Ming Chu, Yue-Ping Jiang · Journal of Inequalities and Applications · 2009

Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially affirmative answer to an open problem posed by B.-N.Guo and F.Qi. Several inequalities for the geometric means of natural numbers are established.

Cited by 34Open access

Generalizations of Convex Functions

2021

Convex functions have convex Lebesgue sets. So convex functions are quasi-convex one. The concave function is logarithmically concave. In addition, the logarithmically convex function is quasi-convex, and the logarithmically concave function is quasi-concave. Many properties of convex functions remain valid or have analogues for quasi-convex functions, for example, the analogue of Jensen's inequality, which characterizes quasi-convexity. Obviously, the product and the scale change preserve the logarithmic convexity and logarithmic concavity. Simple examples show that the sum of logarithmically concave functions may not be a logarithmically concave function. But the sum of logarithmically convex functions is a logarithmically convex function. If the function f is lower semicontinuous, then its strict quasi-convexity implies the usual quasi-convexity.

Cited by 0

Karamata inequality for logarithmically convex functions

С.И. Калинин · Advanced science. · 2020

В статье воспроизводится неравенство Караматы для выпуклой функции, вводится в рассмотрение аналог данного неравенства – неравенство Караматы для логарифмически выпуклой функции (логарифмически вогнутой функции). Автор предлагает читателю также осмысление неравенства Караматы для логарифмически выпуклой или логарифмически вогнутой функций в строгом (нестрогом) смысле для кортежей действительных чисел из двух элементов. Отдельное внимание уделяется вопросу решения уравнений специального вида, порождаемых строго логарифмически выпуклыми или строго логарифмически вогнутыми функциями. Формулируется теорема, описывающая новый метод решения таких уравнений. В качестве иллюстрации применения данной теоремы в работе обсуждаются решения уравнений, которые читателю могут быть потенциально известны из различных литературных источников. The article reproduces the Karamata inequality for a convex function, and introduces an analog of this inequality – the Karamata inequality for a logarithmically convex function (a logarithmically concave function). The author also offers the reader an understanding of the Karamata inequality for logarithmically convex or logarithmically concave functions in the strict (non-strict) sense for tuples of real numbers consisting of two elements. Special attention is paid to the problem of solving special-type equations generated by strictly logarithmically convex or strictly logarithmically concave functions. A theorem describing a new method for solving such equations is formulated. As an illustration of the application of this theorem, the paper discusses solutions of equations that the reader may potentially know from various literature sources

Cited by 0Open access

Some inequalities involving ratios and products of the gamma function

Mridula Garg, Ajay K‎. ‎Sharma, Pratibha Manohar · SHILAP Revista de lepidopterología · 2013

<em> </em>In this paper, we establish some generalized inequalities for the gamma function using the properties of logarithmically convex/concave functions.

Cited by 0Open access

A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators

Muhammad Tariq, Sotiris K. Ntouyas, Asif Ali Shaikh · Mathematics · 2023

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α,m)-convex functions, (α,m)-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ,m)-convex functions, m-harmonic harmonically convex functions, (s,r)-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, (α,m)-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable ϕ-convex functions, MT-convex functions, (s,m)-convex functions, p-convex functions, h-convex functions, σ-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, σ,s-convex functions, modified (p,h)-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, (α,h−m)−p-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, (k,s)-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator.

Cited by 14Open access

Sparsification of sums with respect to convex cones

James Saunderson · arXiv (Cornell University) · 2025

Let $x_1,x_2,\ldots,x_m$ be elements of a convex cone $K$ such that their sum, $e$, is in the relative interior of $K$. An $ε$-sparsification of the sum involves taking a subset of the $x_i$ and reweighting them by positive scalars, so that the resulting sum is $ε$-close to $e$, where error is measured in a relative sense with respect to the order induced by $K$. This generalizes the influential spectral sparsification model for sums of positive semidefinite matrices. This paper introduces and studies the sparsification function of a convex cone, which measures, in the worst case over all possible sums from the cone, the smallest size of an $ε$-sparsifier. The linear-sized spectral sparsification theorem of Batson, Spielman, and Srivastava can be viewed as a bound on the sparsification function of the cone of positive semidefinite matrices. This result is generalized to a family of convex cones (including all hyperbolicity cones) that admit a $ν$-logarithmically homogeneous self-concordant barrier with certain additional properties. For these cones, the sparsification function is bounded above by $\lceil4ν/ε^2\rceil$. For general convex cones that only admit an ordinary $ν$-logarithmically homogeneous self-concordant barrier, the sparsification function is bounded above by $\lceil(4ν/ε)^2\rceil$. Furthermore, the paper explores how sparsification functions interact with various convex geometric operations (such as conic lifts), and describes implications of sparsification with respect to cones for certain conic optimization problems.

Cited by 0Open access

Sparsification of sums with respect to convex cones

James Saunderson · ArXiv.org · 2025

Let $x_1,x_2,\ldots,x_m$ be elements of a convex cone $K$ such that their sum, $e$, is in the relative interior of $K$. An $ε$-sparsification of the sum involves taking a subset of the $x_i$ and reweighting them by positive scalars, so that the resulting sum is $ε$-close to $e$, where error is measured in a relative sense with respect to the order induced by $K$. This generalizes the influential spectral sparsification model for sums of positive semidefinite matrices. This paper introduces and studies the sparsification function of a convex cone, which measures, in the worst case over all possible sums from the cone, the smallest size of an $ε$-sparsifier. The linear-sized spectral sparsification theorem of Batson, Spielman, and Srivastava can be viewed as a bound on the sparsification function of the cone of positive semidefinite matrices. This result is generalized to a family of convex cones (including all hyperbolicity cones) that admit a $ν$-logarithmically homogeneous self-concordant barrier with certain additional properties. For these cones, the sparsification function is bounded above by $\lceil4ν/ε^2\rceil$. For general convex cones that only admit an ordinary $ν$-logarithmically homogeneous self-concordant barrier, the sparsification function is bounded above by $\lceil(4ν/ε)^2\rceil$. Furthermore, the paper explores how sparsification functions interact with various convex geometric operations (such as conic lifts), and describes implications of sparsification with respect to cones for certain conic optimization problems.

Cited by 0Open access

The Function (b^x-a^x)/x: Logarithmic Convexity

Feng Qi, Bai‐Ni Guo · Victoria University Research Repository (Victoria University) · 2008

In the article, logarithmically convex properties of the function (b^x-a^x)/x for x not equal to 0 and positive numbers a and b are investigated, several equalities and inequalities of Bernoulli’s numbers and their generalizations are established, and some related problems and applications are introduced.

Cited by 2

On the second hankel determinant for the kth-root transform of analytic functions

Najla M. Alarifi, Rosihan M. Ali, V. Ravichandran · Filomat · 2017

Let f be a normalized analytic function in the open unit disk of the complex plane satisfying zf'(z)/f(z) is subordinate to a given analytic function ?. A sharp bound is obtained for the second Hankel determinant of the kth-root transform z[f(zk)/zk]1/k. Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes, which include the class of ?-convex functions and ?-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function ?, and thus connect with earlier known results for particular choices of ?.

Cited by 13Open access

Log versus linear timing in human temporal bisection: A signal detection theory study.

Jérémie Jozefowiez, Claire Gaudichon, Francis Mekkass et al. · Journal of Experimental Psychology Animal Learning and Cognition · 2018

Using signal detection theory, we investigated whether human participants represent time linearly or logarithmically in a bisection task. Participants saw a stimulus 1.0 to 1.5 s in duration, and then judged whether the stimulus duration was closer to 1.0 s or to 1.5 s, and how sure they were of their response. Whereas the mean of the subjective stimulus duration was a linear function of the objective stimulus duration, participants produced remarkably different psychophysical functions-linear for some participants, concave for others, and convex for still others. Hence, the appropriate question might not be whether humans encode time linearly or logarithmically, but for which participants and under which conditions is time encoded linearly, logarithmically, or even exponentially. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

Cited by 12Open access

Bounds for some entropies and special functions

Adina Bărar, Gabriela Raluca Mocanu, Ioan Raşa · Carpathian Journal of Mathematics · 2018

We consider a family of probability distributions depending on a real parameter and including the binomial, Poisson and negative binomial distributions. The corresponding index of coincidence satisfies a Heun differential equation and is a logarithmically convex function. Combining these facts we get bounds for the index of coincidence, and consequently for Renyi and Tsallis entropies of order 2.

Cited by 9Open access

Convex conjugates of analytic functions of logarithmically convex\n functionals

Krzysztof Zajkowski · arXiv (Cornell University) · 2011

Let $f_{\\bf c}(r)=\\sum_{n=0}^\\infty e^{c_n}r^n$ be an analytic function;\n${\\bf c}=(c_n)\\in l_\\infty$. We assume that $r$ is some logarithmically convex\nand lower semicontinuous functional on a locally convex topological space $L$.\nIn this paper we derive a formula on the Legendre-Fenchel transform of a\nfunctional $\\hat{\\lambda}({\\bf c},\\phi)=\\ln f_{\\bf c}(e^{\\lambda(\\phi)})$,\nwhere $\\lambda(\\phi)=\\ln r(\\phi)$ ($\\phi\\in L$). In this manner we generalize\nto the infinite case Theorem 3.1 from \\cite{OZ1}.\n

Cited by 0Open access

Convex conjugates of analytic functions of logarithmically convex functionals

Krzysztof Zajkowski · arXiv (Cornell University) · 2011

Let $f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n$ be an analytic function; ${\bf c}=(c_n)\in l_\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this paper we derive a formula on the Legendre-Fenchel transform of a functional $\hatλ({\bf c},ϕ)=\ln f_{\bf c}(e^{λ(ϕ)})$, where $λ(ϕ)=\ln r(ϕ)$ ($ϕ\in L$). In this manner we generalize to the infinite case Theorem 3.1 from \cite{OZ1}.

Cited by 0Open access

On logarithmically superquadratic functions

Mario Krnić, Hamid Reza Moradi, Mohammad Sababheh · Research Square · 2023

Abstract In this paper, we develop a concept of a logarithmically super\-qua\-dratic function. Such a class of functions is defined via superquadratic functions.These functions possess some superior properties, especially if they take values greater than or equal to one. We prove that they are convex and superadditive in the latter case.In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Furthermore, we derive an external form of the Jenseninequality and the corresponding reverse. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions.All established results are derived via the corresponding relations for superquadratic functions.

Cited by 7Open access

On logarithmically superquadratic functions

Mario Krnić, Hamid Reza Moradi, Mohammad Sababheh · Greater South Information System · 2023

Abstract In this paper, we develop a concept of a logarithmically super\-qua\-dratic function. Such a class of functions is defined via superquadratic functions.These functions possess some superior properties, especially if they take values greater than or equal to one. We prove that they are convex and superadditive in the latter case.In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Furthermore, we derive an external form of the Jenseninequality and the corresponding reverse. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions.All established results are derived via the corresponding relations for superquadratic functions.

Cited by 0Open access

Properties and applications of a function involving exponential functions

Bai‐Ni Guo, Feng Qi · Communications on Pure &amp Applied Analysis · 2009

In the present paper, we give necessary and sufficient conditions for the elementary function $ q_{\alpha,\beta}(t)=\frac{e^{-\alpha t}-e^{-\beta t}}{1-e^{-t}},$ if $t\ne 0$ or $q_{\alpha,\beta}(t)=\beta-\alpha,$ if $t=0$ to be monotonic or logarithmically convex on $(-\infty,\infty)$, $(-\infty,0)$ or $(0,\infty)$ respectively, where $\alpha$ and $\beta$ are real numbers and satisfy $\alpha\ne\beta$ and $(\alpha,\beta)$ ∉ {$(0,1),(1,0)$}. Utilizing the monotonicity of $q_{\alpha,\beta}(t)$ on $(0,\infty)$, we derive necessary and sufficient conditions for the function $H_{a,b;c}(x)=(x+c)^{b-a} \frac{\Gamma(x+a)}{\Gamma(x+b)}$, its $q$-analogue, and ratios of the gamma or $q$-gamma functions to be logarithmically completely monotonic, where $a,b,c$ are real numbers and $x\in (-\min$ {$a,b,c$},$\infty)$.

Cited by 37