函数概念与基本初等函数

1.本单元知识点函数是描述客观世界变化规律的重要数学模型.函数内涵丰富、思想深刻、应用广泛,是高中数学的核心知识与关键内容.基本初等函数尽管简单,但非常根本,也能大致满足描摹现实世界的需要.本单元学习重点包括:函数的概念及表示,函数的定义域与值域,函数的单调性与最值、函数的奇偶性,幂运算与对数运算,指数函数、对数函数与幂函数的概念和性质,函数零点与方程根的联     

初等与非初等函数的关系

首先对现行教材中初等函数的定义提出了商讨意见,讨论了高等数学教材中出现的形式上的非初等函数与初等函数的关系,并通过一些有代表性的例子加以说明.     

关于亚纯函数与其导函数分担小函数的一些结果

研究了亚纯函数与其导函数权分担一个小函数的唯一性问题,所得到的定理改进与推广了张继龙、杨连中(2007,2009),Li J D(2015)的结果,并回答了K.W.Yu(2003)提出的问题.     

三角函数的推广

利用函数方程定义仿正切函数类,再用仿正切函数定义仿正、余弦函数类,并对这些函数类进行研究.所定义的函数类中的函数具有通常三角函数的性质及运算规律,故而可将其作为三角函数的某种推广.通过实例证实.了通常的正切函数其实是仿正切函数的一种解析延拓.     

有关单位根集合上解析函数的一些计算和推广

在文献[1]中,Kazuo Habiro给出了单位根集上解析函数的概念.这是一类特殊的含一个变量x的级数.此类级数收敛当且仅当x是单位根.Habiro通过特殊方法构造了这种单位根集合上解析函数的导函数与在单位根处的泰勒展开.但是,对这些函数的导函数与泰勒展开文献[1]并未给出详细证明.因此,将通过详细的计算来研究这类解析函数的导函数和泰勒展开.引入比单位根集合更广的"集合系统"概念,把单位根集合上的解析函数推广到集合系统上.而且这些推广的解析函数仍能用类似的方式定义导函数和泰勒展开.     

一类奇异积分算子的加权估计

假设A是一个Young函数,MD#为广义sharp极大函数.本文首先引进了LA-Hrmander条件,对于满足LA-Hrmander条件的算子T,得到了与T相关的广义sharp极大函数的估计.然后,再利用该估计得到了算子T的加权Lp范数被Hardy-Littlewood极大函数和与Aˉ相关的极大函数的加权Lp范数所控制(0相似文献   

关于Bloch函数的性质

研究了Bloch函数族B中的两个子族Bα与B0,给出了它们的函数之间的关系;作为应用建立了Bα与B0中函数的偏差定理,从而刻画了Bα与B0中函数的有关性质.     

含有解析函数与亚纯函数的一些不等式

指出文献 [2 ]和文献 [4]中的错误 ,并且得到含有单位圆盘内接解析函数与亚纯函数的几个不等式 .     

一个关于广义完全数的数论函数方程

本文运用初等方法,讨论了一个含有约数函数、约数和函数与Euler函数的数论函数方程,给出了该方程的全部偶数解,并且解决了一个有关广义完全数的问题.     

绝对值函数的一致光滑逼近函数

绝对值函数是一个非光滑函数,研究了绝对值函数的光滑逼近函数.给出了绝对值函数的上方一致光滑逼近函数和下方一致光滑逼近函数,分别研究了其性质,并通过图像展示了逼近效果.     

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

In this paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums. They establish three new inequalities involving finitely many exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers. They also survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finitely many gamma functions, present complete monotonicity of a linear combination of finitely many trigamma functions, construct a new ratio of finitely many gamma functions, derive monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finitely many gamma functions. Finally, they suggest two linear combinations of finitely many trigamma functions and two ratios of finitely many gamma functions to be investigated.     

函数变换缩小数据级比偏差效果的比较原则

针对缩小序列级比偏差的单调变换,研究了在函数的和变换与每一个变换缩小级比偏差效果优劣的比较原则基础上证明了两函数之正线性和变换缩小级比偏差效果介于这两个函数缩小级比偏差效果之间.因此如果两函数之正线性和变换不比任何一个单独变换效果差,其实质是两函数商的比值是常数,即两函数变换缩小级比偏差效果一样好,并通过实例验证了结论.     

Cauchy中值定理与Leibniz公式的推广

引入辅助函数的方法可将Cauchy中值定理推广到高阶形式,即两函数n阶Taylor展开误差的比值等于在某点两函数(n+1)阶导数比值的形式;用数学归纳法可将Leibniz公式中函数的个数推广至任意有限多个.     

Competitive Boolean function evaluation: Beyond monotonicity, and the symmetric case

We study the extremal competitive ratio of Boolean function evaluation. We provide the first non-trivial lower and upper bounds for classes of Boolean functions which are not included in the class of monotone Boolean functions. For the particular case of symmetric functions our bounds are matching and we exactly characterize the best possible competitiveness achievable by a deterministic algorithm. Our upper bound is obtained by a simple polynomial time algorithm.     

Two new asymptotic expansions of the ratio of two gamma functions

Formal expansions, giving as particular cases semiasymptotic expansions, of the ratio of two gamma functions are obtained.     

A new upper bound in the second Kershaw's double inequality and its generalizations

In the paper, a new upper bound in the second Kershaw's double inequality involving ratio of gamma functions is established, and, as generalizations of the second Kershaw's double inequality, the divided differences of the psi and polygamma functions are bounded.     

On ratio block sequences with extreme distribution function

Distribution functions of ratio block sequences formed from sequences of positive integers are investigated in the paper. We characterize the case when the set of all distribution functions of a ratio block sequence contains c ₀, the greatest possible distribution function. Presented results complete some previously published results. Supported by grant MSM 6198898701, and VEGA no. 1/4006/07.     

An Expression for the Superposition of General Algebraic Functions in Terms of Hypergeometric Series

We show that the superposition of general algebraic functions is representable as a ratio of hypergeometric series.     

Elementary Proof of MacMahon's Conjecture

Major Percy A. MacMahon's first paper on plane partitions [4] included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series [1] and Macdonald employing his knowledge of symmetric functions [3]. The purpose of this paper is to simplify Macdonald's proof by providing a direct, inductive proof of his formula which expresses the sum of Schur functions whose partitions fit inside a rectangular box as a ratio of determinants.     

A logarithmically completely monotonic function involving the ratio of gamma functions

In the paper, the authors concisely survey and review some functionsinvolving the gamma function and its various ratios, simply state theirlogarithmically complete monotonicity and related results, and find necessaryand sufficient conditions for a new function involving the ratio of twogamma functions and originating from the coding gain to be logarithmicallycompletely monotonic.