تابع یکنوای عملگری و تحدب نرم مشتق‌های آن

فرض کنید f یک تابع یکنوای عملگری روی ( infin;,0) و a یک عملگرمثبت وارون پذیر روی فضای هیلبرت h باشد.نشان می دهیم اگر |||.||| یک نرم یکانی پایا باشد، آن گاه برای هر عدد صحیح مثبت n،ثابت می کنیم تابع∥(.)f(n)∥روی مجموعه ی همه ی عملگرهای مثبت وارون پذیر درb(h) شبه‌محدب است. و همچنین نشان می دهیم :که این یک تظریف از نتیجه معروف زیر است:که در آن a یک عدد حقیقی مثبت و a,b le;a1h .ما در این مقاله برخی تقریب‌ها از طرف راست نامساویهای نوع هرمیت -هاداماردکه شامل توابع مشتق‌پذیرندونرم نگاشت‌های القاءشده توسط آن‌ها روی مجموعه تمام عملگرهای خودالحاق، محدب یا شبه‌محدب یاs محدب هستند، به‌دست می آوریم.

Operator Monotone Functions and Convexity of Its Derivatives Norms

Introduction Given the important role convex and quasiconvex functions play in many areas of mathematics and especially in optimization, one of the inequalities that has attracted the attention of many mathematicians in recent decades is HermitHadamard rsquo;s famous inequality. Significant generalizations and refinements have been obtained for this inequality in a diverse variety of convexity, including convex operator functions of self adjoint operators on Hilbert spaces, matrix functions, quasiconvex, sconvex and logconvex functions.In this paper, we generalize this inequality to differentiable functions whose norm of their derivatives are convex functions.Results and discussionIn this paper, we consider differentiable mappings which norm of the induced maps by them on the set of self adjoint operators is convex, quasi convex or sconvex. We show that if is an operator monotone function on , is a strictly positive operator and a unitarily invariant norm, then for all positive integers . We also prove that is a quasiconvex function on the set of all strictly positive operators in B(H). Examples and applications for particular cases of interest are also illustrated. Finally, an error estimate for the Simpson formula is addressed.ConclusionThe following conclusions were drawn from this research.As an important application of the results in this paper, we find bounds for in terms of , which is one of the central problems in perturbation theory.We establish some estimates of the right hand side of some HermiteHadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasiconvex or s convex../files/site1/files/71/5.pdf