A New Formula for the Natural Logarithm of a Natural Number

For every natural number $T,$ we write \Ln T as a series, generalizing the known series for \Ln 2.Comment: 4 page

A Non explicit counterexample to a problem of quasi-normality

In 1986, S.Y. Li and H.Xie proved the following theorem:Let k>=2 and let F be a family of functions meromorphic in some domain D, all of whose zeros are of multiplicity at least k. Then F is normal if and only if the family F_k={f^(k)/(1+|f^k+1|):f in F} is locally uniformly bounded in D. Here we give, in the case k=2, a counterexample to show that if the condition on the multiplicities of the zeros is omitted, then the local uniform boundedness of F_2 does not imply even quasi-normality. In addition, we give a simpler proof for the Li-Xie Theorem that does not use Nevanlinna Theory which was used in the original proof

Some counterexamples on the behaviour of real-valued functions and their derivatives

We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict minimum doesn't have to be decreasing to the left nor increasing to the right of the minimum, we present a function whose derivative is discontinuous at one point and has a strict minimum at this point (i.e. it oscillates only in one direction), we compare several definitions of inflection point, and we discuss a general version of the theorem on the derivative of inverse functions where continuity of the inverse function is assumed merely at one point

An Extension of a Normality Result by Carath\'eodory

We extend the fundamental normality test due to Carath\'eodory in the sense of shared functions.Comment: 6 page

On the growth of real functions and their derivatives

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot \log_q x \cdot f^{(k)}(x)}{1+|f(x)|^\alpha}\le 0$$ holds.Comment: 10 page

Creating Limit Functions By The Pang-Zalcman Lemma

In this paper we calculate the collection of limit functions obtained by applying an extension of Zalcman's Lemma, due to X. C. Pang, to the non-normal family $\left\{f(nz):n\in\mathbb{N}\right\}$ in $\mathbb{C}$, where $f=Re^P$. Here $R$ and $P$ are an arbitrary rational function and a polynomial, respectively, where $P$ is a non-constant polnomial

A criterion of normality based on a single holomorphic function II

In this paper, we continue to discuss normality based on a single\linebreak holomorphic function. We obtain the following result. Let \CF be a family of functions holomorphic on a domain $D\subset\mathbb C$. Let $k\ge2$ be an integer and let $h(\not\equiv0)$ be a holomorphic function on $D$, such that $h(z)$ has no common zeros with any f\in\CF. Assume also that the following two conditions hold for every f\in\CF:\linebreak %{enumerate} [(a)] (a) $f(z)=0\Longrightarrow f'(z)=h(z)$ and %[(b)] (b) $f'(z)=h(z)\Longrightarrow|f^{(k)}(z)|\le c$, where $c$ is a constant. Then \CF is normal on $D$. %{enumerate} A geometrical approach is used to arrive at the result which significantly improves the previous results of the authors, \textit{A criterion of normality based on a single holomorphic function}, Acta Math. Sinica, English Series (1) \textbf{27} (2011), 141--154 and of Chang, Fang, and Zalcman, \textit{Normal families of holomorphic functions}, Illinois Math. J. (1) \textbf{48} (2004), 319--337. We also deal with two other similar criterions of normality. Our results are shown to be sharp

A note on spherical derivatives and normal families

We show that a family of meromorphic functions in the unit disk \dk whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each $f$ meromorphic in \dk we have \inf_{z\in\dk} f^#(z)\le \frac{1}{2}$where$f^#$denotes the spherical derivative of$f$.Comment: 12 page

Differential inequalities and quasi-normal families

We show that a family ${\cal F}$ of meromorphic functions in a domain $D$ satisfying \frac{|f^{(k)}|}{1+|f^{(j)}|^\alpha}(z)\ge C \qquad \mbox{for all} z\in D \mbox{and all} f\in {\cal F} (where $k$ and $j$ are integers with $k>j\ge 0$ and $C>0$, $\alpha>1$ are real numbers) is quasi-normal. Furthermore, if all functions in ${\cal F}$ are holomorphic, the order of quasi-normality of ${\cal F}$ is at most $j-1$. The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality

Differential inequality of the second derivative that leads to normality

Let F be a family of functions meromorphic in a domain D. If {|f|/(1+|f|^3):f in F} is locally uniformly bounded away from zero, then F is normal