اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدعلى عوامل أندوس للدوال المنحنية والمنحنية العكسية
On Andos inequalities for convex and concave functions
255
0
0.0
(
0
)
اسأل ChatGPT حول البحث
بالنسبة للمصفوفات الثنائية الموجبة النقطية $A$ و $B$، أثبت Ando و Zhan الإنحدارات $||| f(A)+f(B) ||| ge ||| f(A+B) |||$ و $||| g(A)+g(B) ||| le ||| g(A+B) |||$، لأي نوع من الأنظمة الثابتة المتحيزة، و لأي دالة تحفظ الأولوية السالبة على $[0,infty)$ بإخلاء دالة $g$. وقد تم توسيع هذه الإنحدارات حديثا إلى الدوال السالبة الغير موجبة والدوال المثبتة الغير منفية عن طريق Bourin و Uchiyama و Kosem على التوالي. في هذا البحث نناقش السؤال المرتبط بذلك حول ما إذا كانت الإنحدارات $||| f(A)-f(B) ||| le ||| f(|A-B|) |||$، و $||| g(A)-g(B) ||| ge ||| g(|A-B|) |||$، التي تم الحصول عليها من قبل Ando لدالة تحفظ الأولوية $f$ مع إخلاء $g$، يوجد لها توسيع مشابه إلى الدوال السالبة الغير موجبة والدوال المثبتة الغير منفية؟ ونجيب على هذا السؤال بشكل سلبي للمصفوفات العامة، وبشكل موجب للحالة الخاصة عندما $Age ||B||$. في طور هذا العمل، نقدم المفهوم الجديد للتحكم المغلوب بين الطيفين لمصفوفتين الهيرميتيتين، حيث يكون $Y$ نفسه مصفوفة هيرميتية، ونبرهن خاصية من هذا العلاق التي تسمح لتعزيز نتائج Bourin-Uchiyama و Kosem المذكورة أعلاه.
For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $||| f(A)+f(B) ||| ge ||| f(A+B) |||$ and $||| g(A)+g(B) ||| le ||| g(A+B) |||$, for any unitarily invariant norm, and for any non-negative operator monotone $f$ on $[0,infty)$ with inverse function $g$. These inequalities have very recently been generalised to non-negative concave functions $f$ and non-negative convex functions $g$, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities $||| f(A)-f(B) ||| le ||| f(|A-B|) |||$, and $||| g(A)-g(B) ||| ge ||| g(|A-B|) |||$, obtained by Ando, for operator monotone $f$ with inverse $g$, also have a similar generalisation to non-negative concave $f$ and convex $g$. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when $Age ||B||$. In the course of this work, we introduce the novel notion of $Y$-dominated majorisation between the spectra of two Hermitian matrices, where $Y$ is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above.
قيم البحث
اقرأ أيضاً
Sub-additive and super-additive inequalities for concave and convex functions have been generalized to the case of matrices by several authors over a period of time. These lead to some interesting inequalities for matrices, which in some cases coinci
de with, and in other cases are at variance with the corresponding inequalities for real numbers. We survey some of these matrix inequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectra of two Hermitian matrices $B$ and $C$, dominated by a third Hermitian matrix $A$. Based on an explicit formula for the gradient of the sum of the $k$ largest eigenvalues of a Hermitian matrix, we show that under certain conditions dominated majorization reduces to a linear majorization-like relation between the diagonal elements of $B$ and $C$ in a certain basis. We use this notion as a tool to give new, elementary proofs for the sub-additivity inequality for non-negative concave functions first proved by Bourin and Uchiyama and the corresponding super-additivity inequality for non-negative convex functions first proven by Kosem. Finally, we present counterexamples to some conjectures that Andos inequality for operator convex functions could more generally hold, e.g. for ordinary convex, non-negative functions.
In this paper, we introduce the concept of operator geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results.
In this paper, we introduce the concept of operator arithmetic-geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for opera
tors which give some refinements of previous results. Moreover, some unitarily invariant norm inequalities are established.
Convexification based on convex envelopes is ubiquitous in the non-linear optimization literature. Thanks to considerable efforts of the optimization community for decades, we are able to compute the convex envelopes of a considerable number of funct
ions that appear in practice, and thus obtain tight and tractable approximations to challenging problems. We contribute to this line of work by considering a family of functions that, to the best of our knowledge, has not been considered before in the literature. We call this family ray-concave functions. We show sufficient conditions that allow us to easily compute closed-form expressions for the convex envelope of ray-concave functions over arbitrary polytopes. With these tools, we are able to provide new perspectives to previously known convex envelopes and derive a previously unknown convex envelope for a function that arises in probability contexts.
Affine invariant points and maps for sets were introduced by Grunbaum to study the symmetry structure of convex sets. We extend these notions to a functional setting. The role of symmetry of the set is now taken by evenness of the function. We show t
hat among the examples for affine invariant points are the classical center of gravity of a log-concave function and its Santalo point. We also show that the recently introduced floating functions and the John- and Lowner functions are examples of affine invariant maps. Their centers provide new examples of affine invariant points for log-concave functions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
