毕业论文-Jensen不等式的推广05418.doc

本科毕业论文（设计）题 目： jensen不等式的推广院（系）专业： 数学系（数学与应用数学）学生姓名： 冯德文 学 号： 2003701107 导师（职称）： 杨慧章 (助教) 日 期： 2012年6月 红河学院本科毕业论文(设计)摘 要凸函数是一种性质特殊的函数，而凸函数的jensen 不等式是一个很重要的不等式，由它可推出一系列不等式，而凸函数的构造也有其妙处。为使其更广泛应用于不等式的证明，本文利用凸函数的性质对jensen不等式进行了推广，得到几个重要的积分不等式并进行了证明。关键词：凸函数 ； 积分 abstractthe convex function is one function with special properties, but the jensen inequality of convex function is a very important inequality. according to the function, we can evolve a series of inequalities, and use it more easily to prove some important inequalities, but the convex function structures also have their advantages, in order to make good use of proving inequalities widely, in this paper, we use the properties of convex function to expand the jensen inequality, obtain several important integral inequalities and give the proof of them.key word：convex function；integralii目 录绪论11 预备知识21.1 凸函数21.2 jensen不等式22 jensen不等式的推广42.1 积分型jensen不等式42.2 其它积分不等式52.3 应用8结论10感谢信11参考文献12绪论绪 论不等式是研究分析数学的重要工具，在高等数学中我们要用到各种形式的不等式。本文主要利用凸函数的定义及性质去证明不等式。其关键是寻找合适的凸函数,若不能直接找出,则对不等式进行适当的变形,从而达到证明不等式的目的。本文内容安排如下：第一章 预备知识。先介绍凸函数的定义及充要条件，再给出凸函数的jensen不等式及其证明。第二章 jensen不等式的推广。先利用凸函数的定义及性质把前一章给出的jensen不等式推广到积分形式，并给出证明。再由前章给出的知识以及积分型的jensen不等式推出几个重要积分不等式并进行证明。最后给出两个例子介绍它们的应用。 11预备知识1 预备知识1.1 凸函数定义 设为定义在区间上的函数，若对上的任意两点和任意实数总有则称为上的凸函数。反之，如果总有则称为上的凹函数。定理1 设为上的可导函数，则为上的凸函数的充要条件是， 或 对上的任意两点，有 1.2 jensen不等式定理2 (jensen不等式) 为区间上的凸函数，则对任意，且，有 (1-1)2再把上式两端分别相加，得由 及，上式变为=即注：当时，有，则(1-1)式变为 (1-2) 3结论2 jensen不等式的推广2.1 积分型jensen不等式命题1 若在区间上连续，处处2阶可导且，则有 (2-1)证法：把区间等分，把代入(1-2)式，有 即 因为，在上连续，当时，有 所以2.2 其它积分不等式命题2 若在连续，则 (2-3)证明：设，则，所以为凸函数。由命题1可得 即 所以 注：命题2为命题1的一般形式，相当于命题一中的。因为为凹函数，所以符号相反。命题3 若在区间连续且，则 (2-4)证法一：设，则。因为，所以，即为凸函数。根据命题1有 即 结论得证。 注：命题3由命题1所得，相当于。证法二：把 等分，分点为。因为算术平均值大于调和平均值，所以有 =由有 令，取极限得 结论得证。命题5 设在连续，且则有证：因为函数 为凸函数，由jensen不等式有 =。 =综上可得 注：上式为均值不等式。2.3 应用例1 证明。证：令因为且的连续性，所以由jensen不等式有 = = =。结 论凸函数是一个传统研究课程，具有广泛的实际背景和应用价值。对凸函数性质的探讨是一个重要的研究方向。 本文凸函数jensen不等式的应用仅仅是限于一元函数而言，可将其推广到多元函数，将空间扩充到凸集的范围，这些类似定理和结论以及相关应用有待一步研究。9红河学院本科毕业论文(设计)感谢信在毕业论文完成之际,向给予我帮助和指导的各位老师和同学表示衷心的感谢!首先,我要感谢我的论文导师杨慧章老师,因为有她耐心的指导、鼓励和帮助我才顺利完成我的毕业论文。借此机会我向数学系老师表示衷心的感谢,感谢他们四年来的精心指导和培养。10参考文献1林玎,刘伟.jensen不等式的几个推论及其应用j.吉林师范大学学报,2003,8(3)：28-232刘飞燕.jensen不等式及其应用j.云南民族大学学报,2003,7(3)：148-1513徐伟.积分形jensen不等式的巧用j.高等数学研究,2002,6(4)：15-164陈欣.关于jensen不等式的应用j.武汉工业学院学报,2005,9(3)：113-1155刘鸿雁.由jensen不等式导出某些重要不等式 j.成都大学学报 ,2003,12(4)：32-356尚亚东,游淑军.凸函数及其在不等式证明中的应用j.广州大学学报,2005,2(1):1-6. g an employment tribunal claimemployment tribunals sort out disagreements between employers and employees.you may need to make a claim to an employment tribunal if: you dont agree with the disciplinary action your employer has taken against you your employer dismisses you and you think that you have been dismissed unfairly.for more information about dismissal and unfair dismissal, seedismissal.you can make a claim to an employment tribunal, even if you haventappealedagainst the disciplinary action your employer has taken against you. however, if you win your case, the tribunal may reduce any compensation awarded to you as a result of your failure to appeal.remember that in most cases you must make an application to an employment tribunal within three months of the date when the event you are complaining about happened. if your application is received after this time limit, the tribunal will not usually accept it.if you are worried about how the time limits apply to you, take advice from one of the organisations listed underfurther help.employment tribunals are less formal than some other courts, but it is still a legal process and you will need to give evidence under an oath or affirmation.most people find making a claim to an employment tribunal challenging. if you are thinking about making a claim to an employment tribunal, you should get help straight away from one of the organisations listed underfurther help.if you are being represented by a solicitor at the tribunal, they may ask you to sign an agreement where you pay their fee out of your compensation if you win the case. this is known as adamages-based agreement. in england and wales, your solicitor cant charge you more than 35% of your compensation if you win the case.if you are thinking about signing up for a damages-based agreement, you should make sure youre clear about the terms of the agreement. it might be best to get advice from an experienced adviser, for example, at a citizens advice bureau. to find your nearest cab, including those that give advice by e-mail, click onnearest cab.for more information about making a claim to an employment tribunal, seeemployment tribunals.the (lack of) air up there watch mcayman islands-based webb, the head of fifas anti-racism taskforce, is in london for the football associations 150th anniversary celebrations and will attend citys premier league match at chelsea on sunday.i am going to be at the match tomorrow and i have asked to meet yaya toure, he told bbc sport.for me its about how he felt and i would like to speak to him first to find out what his experience was.uefa hasopened disciplinary proceedings against cskafor the racist behaviour of their fans duringcitys 2-1 win.michel platini, president of european footballs governing body, has also ordered an immediate investigation into the referees actions.cska said they were surprised and disappointed by toures complaint. in a statement the russian side added: we found no racist insults from fans of cska.age has reached the end of the beginning of a word. may be guilty in his seems to passing a lot of different life became the appearance of the same day; may be back in the past, to oneself the paranoid weird belief disillusionment, these days, my mind has been very messy, in my mind constantly. always feel oneself should go to do something, or write something. twenty years of life trajectory deeply shallow, suddenly feel something, do it.一字开头的年龄已经到了尾声。或许是愧疚于自己似乎把转瞬即逝的很多个不同的日子过成了同一天的样子；或许是追溯过去，对自己那些近乎偏执的怪异信念的醒悟，这些天以来，思绪一直很凌乱，在脑海中不断纠缠。总觉得自己自己似乎应该去做点什么，或者写点什么。二十年的人生轨迹深深浅浅，突然就感觉到有些事情，非做不可了。the end of our life, and can meet many things really do?而穷尽我们的一生，又能遇到多少事情是真正地非做不可？during my childhood, think lucky money and new clothes are necessary for new year, but as the advance of the age, will be more and more found that those things are optional; junior high school, thought to have a crush on just means that the real growth, but over the past three years later, his writing of alumni in peace, suddenly found that isnt really grow up, it seems is not so important; then in high school, think dont want to give vent to out your inner voice can be in the high school children of the feelings in a period, but was eventually infarction when graduation party in the throat, later again stood on the pitch he has sweat profusely, looked at his thrown a basketball hoops, suddenly found himself has already cant remember his appearance.童年时，觉得压岁钱和新衣服是过年必备，但是随着年龄的推进，会越来越发现，那些东西根本就可有可无；初中时，以为要有一场暗恋才意味着真正的成长，但三年过去后，自己心平气和的写同学录的时候，突然就发现是不是真正的成长了，好像并没有那么重要了；然后到了高中，觉得非要吐露出自己的心声才能为高中生涯里的懵懂情愫划上一个句点，但毕业晚会的时候最终还是被梗塞在了咽喉，后来再次站在他曾经挥汗如雨的球场，看着他投过篮球的球框时，突然间发现自己已经想不起他的容颜。originally, this world, can produce a chemical reaction to an event, in addition to resolutely, have to do, and time.原来，这个世界上，对某个事件能产生化学反应的，除了非做不可的坚决，还有，时间。a persons time, your ideas are always special to clear. want, want, line is clear, as if nothing could shake his. also once seemed to be determined to do something, but more often is he backed out at last. dislike his cowardice, finally found that there are a lot of love, there are a lot of miss, like shadow really have been doomed. those who do, just green years oneself give oneself an arm injection, or is a self-righteous spiritual.一个人的时候，自己的想法总是特别地清晰。想要的，不想要的，界限明确，好像没有什么可以撼动自己。也曾经好像已经下定了决心去做某件事，但更多的时候是最后又打起了退堂鼓。嫌恶过自己的怯懦，最终却发现有很多缘分，有很多错过，好像冥冥之中真的已经注定。那些曾经所谓的非做不可，只是青葱年华里自己给自己注射的一支强心剂，或者说，是自以为是的精神寄托罢了。at the moment, the sky is dark, the air is fresh factor after just rained. suddenly thought of blue plaid shirt; those were broken into various shapes of stationery; from the corner at the beginning of deep friendship; have declared the end of the encounter that havent start planning. those years, those days of do, finally, like youth, will end in our life.此刻，天空是阴暗的，空气里有着刚下过雨之后的清新因子。突然想到那件蓝格子衬衫；那些被折成各种各样形状的信纸；那段从街角深巷伊始的友谊；还有那场还没有开始就宣告了终结的邂逅计划那些年那些天的非做不可，终于和青春一样，都将在我们的人生中谢幕。baumgartner the disappointing news: mission aborted. r plays an important role in this mission. starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. the balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. it will climb higher than the tip of mount everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. as he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.athe supersonic descent could happen as early as sunda.the weathethe balloon will slowly drift to the edge of space at 120,000 feet ( then, i would assume, he will slowly step out onto something resembling an olympic diving platform.below, the earth becomes the concrete bottom of a swimming pool that he wants to land on, but not too hard. still, hell be traveling fast, so despite the distance, it will not be like diving into the deep end