fa تابع یکنوای عملگری و تحدب نرم مشتق‌های آن جبر alg مقاله مستقل Original Manuscript <p><span style="font-family: verdana;">فرض کنید f یک تابع یکنوای عملگری روی (&infin;,0) و A یک عملگرمثبت وارون پذیر روی فضای هیلبرت H باشد.نشان می&shy;دهیم اگر |||.||| یک نرم یکانی پایا باشد، آن&shy;گاه برای هر عدد صحیح مثبت n،</span></p> <div style="text-align: center;"><span style="font-family: verdana;"><img alt="" height="20" src="data:image/png;base64,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" width="146" ></span></div> <p><span style="font-family: verdana;">ثابت می&shy;کنیم تابع∥(.)f⁽ⁿ⁾∥روی مجموعه&shy;ی همه&shy;ی عملگرهای مثبت وارون پذیر درB(H) شبه‌محدب است. و همچنین نشان می&shy;دهیم :</span></p> <p style="text-align: center;"><span style="font-family: verdana;"><img alt="" height="26" src="data:image/png;base64,R0lGODlhPgEaAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA8ARUAhQAAAAAAAAAAOgAAZgA6kABmkABmtjoAADoAOjoAZjo6ADo6Ojo6Zjo6kDqQ22YAAGYAOmYAZmY6ZmZmtma2/5A6AJA6OpA6ZpBmAJBmkJCQZpCQ25C2/5Db/7ZmALZmOrZmZrbbkLb/trb/27b//9uQOtuQZtuQkNv///+2Zv/bkP//tv//2wECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikcslsOp9QpEfQAbAqFJalGn1es9vjV8vtOsdhwLRsbrvf8Lh8LvcQUEI0m57HksV+aXxEekMpVIOJiouMjVErEGWFi5NFlYqXHgaOnJ2en24pd0OXg6V9YHumgWyieKCwQyoIqmqbsbhvrqSsULO1mpa9RqdKv1K3vKlEu7mfJaNIzc7UTdPFhg5D0K9H18PCy0kp2kLcSt/iQtPVTSXl7+blvAEDJCoN3UIrDxTb82au1CMhixa/AAgRBSMSD0DDhtRUJCCI6g89ewBMcMGnDwA/f/KUWQw3ss9AjUI47nuAMACikBUFOUzmcN5DgEQ+mjmYkMtN/3k/axbRxCIDCohDTCgACUAiRTNEjZqLVuLWuWlBkSI5twiSJHC2inbUmnGprIkxa5WKqg9p1amvnKYl4gEgxKBGlDKN8tbhqKwOAOeMhKrIigl1ScmE4pWISpjN7sKbnOQgzkSN5+4jTAyLYcTzLBa7lJmQ53/rookGmxg1TLIeQbdpuAuw4JQHEG76UgQECYi8u6jIHSAZ8DA6U6IVyhy2h4FI+BngR+V5sucCNPgrEYCCdMf5RKYZrrvzXgC+gZ8eDY588XCKqyS3sh4sVpuUjaS/3ORPctv5IdVMcClxMJMybaQzBE8Yxfdafgv2swQIITAgwQgVLEDBLnagsP9CBCSssEEJBVzwFADsVMIOgrIY2Bd9ILGnDorRmMYUgycGV0pk+MHkmIs0PYGjazfdZk4yBApUXh6L8dRSADUeaaMhoyjYHIR0vTcOFVdoE09jjx1TxGMwjvQifCYtaUUaMpZ0Jos04nENm/bVKJmPaWoZ4ZMIRbkLjw8aSSBzKJaTQjJydTEogX0lCuODeBaRAkL8xSOXB/7UNo+Yg30lzqBFQETOOogu16YgoLp2YFPLsUpRJnb1SOg6hgbpRKPLAQjpZmzIxYKJcEUqJGeFCYEpfTRZZGQSXBV2qEcRiDCBK5NS4Bs+GDhwwomlTVKaEb4C69cr6i3z4VOVfOv/2HLHXkFTuW7GSiRAv1LU7BPtVmBcgIHxa0iD2xggkD/kERyeGSkA/NqTQSa6rJAgCsWPPQcZwB0FHU6Kk6OTJHyiEVUN3FRuBndTCMfgeMzsJty1RBOZl5x5J6EiF8wXwwzxa+Sb67w0VJR82eoos7LitRKfUHbERzrK2mrEIcBEWcgHbFTCs6Q+/6zPJffN+4zO/j76SaqrJkHmw7gwHQbZb5zKBdtucG1n0fwtouuVHj3gNCPfSZM1XVGiHYvaHfTNh9t57912nd3MDJvdYbcjOS7nilfLHIgzErPik3fuuTMdaoYJWJY3ouLfn6euOixrlLnY4aSLPro4z523BDoRQQAAOw==" width="318" ></span></p> <p><span style="font-family: verdana;">که این یک تظریف از نتیجه معروف زیر است:</span></p> <p style="text-align: center;"><span style="font-family: verdana;"><img alt="" height="21" src="data:image/png;base64,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" width="211" ></span></p> <p><span style="font-family: verdana;">که در آن a یک عدد حقیقی مثبت و A,B&le;a1_H .</span><br> <span style="font-family: verdana;">ما در این مقاله برخی تقریب‌ها از طرف راست نامساویهای نوع هرمیت&ndash;هاداماردکه شامل توابع مشتق‌پذیرندونرم نگاشت‌های القاءشده توسط آن‌ها روی مجموعه تمام عملگرهای خودالحاق، محدب یا شبه‌محدب یاs- محدب هستند، به‌دست می آوریم. &nbsp;</span></p> <strong>Introduction</strong><br> &nbsp;Given the important role convex and quasi-convex 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 Hermit-Hadamard&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, quasi-convex, s-convex and log-convex functions.<br> In this paper, we generalize this inequality to differentiable functions whose norm of their derivatives are convex functions.<br> <strong>Results and discussion</strong><br> In 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 s-convex. We show that if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="4" >&nbsp;is an operator monotone function on <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="38" >, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="9" >&nbsp;is a strictly positive operator and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="33" >&nbsp;a unitarily invariant norm, then&nbsp; <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="152" >for all positive integers <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="8" >. We also prove that<br> <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="59" >&nbsp;is a quasi-convex 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.<br> <strong>Conclusion</strong><br> The following conclusions were drawn from this research. <ul> <li>As an important application of the results in this paper, we find bounds for</li> </ul> <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="87" >&nbsp;in terms of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="54" >, which is one of the central problems in perturbation theory. <ul> <li>We establish some estimates of the right hand side of some Hermite-Hadamard 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, quasi-convex or s- convex.<a href="./files/site1/files/71/5.pdf">./files/site1/files/71/5.pdf</a></li> </ul> نامساوی ارمیت - ادامارد, توابع مشتق پذیر, نرم یکانی پایا, تابع یکنوای عملگری. Hermite-Hadamard inequality, Differentiable functions, Unitarily invariant norms, Operator monotone functions. 43 52 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1022-1&slc_lang=fa&sid=1 Zahra Rahimi Chegeni زهرا رحیمی چگنی Zrahimi2060@yahoo.com 10031947532846004431 10031947532846004431 No دانشگاه پیام نور، گروه ریاضی، تهران amir ghasem ghazanfari امیرقاسم غضنفری ghazanfari.a@lu.ac.ir 10031947532846004432 10031947532846004432 Yes دانشگاه لرستان، گروه ریاضی Kamal Falahi کمال فلاحی fallahi1361@gmail.com 10031947532846004433 10031947532846004433 No دانشگاه پیام نور، گروه ریاضی، تهران